1. This document provides step-by-step instructions for solving various calculus problems including finding zeros, derivatives, integrals, limits, continuity, asymptotes, extrema, and solving differential equations.
2. For each type of problem, it lists the key steps to take in a "You think..." prompt followed by more detailed explanations and formulas.
3. The document is intended as a reference for a student to know the general approach and techniques for multiple calculus problems at a glance.
This document discusses two applications of tangent lines: differentials and linear approximation, and finding the tangent line T(b) at a nearby point b. It explains that the tangent line T(x) at point (a, f(a)) is given by T(x) = f'(a)(x - a) + f(a). The slope f'(a) is identified with the derivative dy/dx. There are two ways to find T(b): directly using T(x), or by finding the differential ΔT = dy and using ΔT + f(a) = T(b).
The document discusses tangent planes to surfaces. A surface is differentiable at a point if it is smooth and has a well-defined tangent plane at that point. The tangent plane approximates the surface near the point of tangency. To find the equation of the tangent plane, we calculate the partial derivatives of the surface function at the point to determine the slopes in the x- and y-directions. These slopes and the point define vectors in the tangent plane, and their cross product gives the normal vector. The equation of the tangent plane is then (z - c) = M(x - a) + L(y - b), where M and L are the partial derivatives and (a, b, c) is the
The document discusses the trace-method for visualizing quadric surfaces. A trace is the intersection of a surface with a plane. For a given surface S, traces parallel to the coordinate planes reveal the shape of S. Specifically, the intersection of S with planes where x=c, y=c, or z=c produce traces perpendicular to the respective axes. The traces of quadric surfaces are conic sections, such as ellipses or circles, which can be used to sketch the overall surface shape.
The document discusses 3D coordinate systems. It explains that a 3D coordinate system adds a z-axis perpendicular to the x and y axes. There are two ways to add the z-axis, resulting in either a right-hand or left-hand system. The right-hand system is used in math and science, while the left-hand system is used in computer graphics. Points in 3D space are identified by ordered triples (x,y,z). Basic graphs in 3D include planes defined by constant equations like x=k, as well as linear equations that define planes.
This document discusses limit laws and theorems for computing limits. It provides 10 limit laws for operations like addition, subtraction, multiplication, division, powers, and roots. It also covers the direct substitution property, that limits can be found by substituting the point into the function if it is defined, and two important theorems: that limits can be compared if one function is always less than or equal to another, and the squeeze/sandwich theorem which allows limits to be found if a function is squeezed between two others with known limits. Examples are provided to illustrate finding limits through algebraic manipulation and using these theorems.
Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard ...Marco Benini
A novel approach to giving an interpretation of logic inside category theory. This work has been developed as part of my sabbatical Marie Curie fellowship in Leeds.
Presented at the Logic Seminar, School of Mathematics, University of Leeds (2012).
The document discusses multi-variable functions and how to visualize them. It defines Rn as the set of n-tuples of real numbers. A n-variable real valued function maps from Rn to the real numbers. Level curves are used to visualize functions of two variables by setting z equal to constants and plotting the resulting equations in the x-y plane. This produces families of curves that depict the shape of the surface. Examples show how different functions produce different level curve shapes like ellipses, hyperbolas or parabolas.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
This document discusses two applications of tangent lines: differentials and linear approximation, and finding the tangent line T(b) at a nearby point b. It explains that the tangent line T(x) at point (a, f(a)) is given by T(x) = f'(a)(x - a) + f(a). The slope f'(a) is identified with the derivative dy/dx. There are two ways to find T(b): directly using T(x), or by finding the differential ΔT = dy and using ΔT + f(a) = T(b).
The document discusses tangent planes to surfaces. A surface is differentiable at a point if it is smooth and has a well-defined tangent plane at that point. The tangent plane approximates the surface near the point of tangency. To find the equation of the tangent plane, we calculate the partial derivatives of the surface function at the point to determine the slopes in the x- and y-directions. These slopes and the point define vectors in the tangent plane, and their cross product gives the normal vector. The equation of the tangent plane is then (z - c) = M(x - a) + L(y - b), where M and L are the partial derivatives and (a, b, c) is the
The document discusses the trace-method for visualizing quadric surfaces. A trace is the intersection of a surface with a plane. For a given surface S, traces parallel to the coordinate planes reveal the shape of S. Specifically, the intersection of S with planes where x=c, y=c, or z=c produce traces perpendicular to the respective axes. The traces of quadric surfaces are conic sections, such as ellipses or circles, which can be used to sketch the overall surface shape.
The document discusses 3D coordinate systems. It explains that a 3D coordinate system adds a z-axis perpendicular to the x and y axes. There are two ways to add the z-axis, resulting in either a right-hand or left-hand system. The right-hand system is used in math and science, while the left-hand system is used in computer graphics. Points in 3D space are identified by ordered triples (x,y,z). Basic graphs in 3D include planes defined by constant equations like x=k, as well as linear equations that define planes.
This document discusses limit laws and theorems for computing limits. It provides 10 limit laws for operations like addition, subtraction, multiplication, division, powers, and roots. It also covers the direct substitution property, that limits can be found by substituting the point into the function if it is defined, and two important theorems: that limits can be compared if one function is always less than or equal to another, and the squeeze/sandwich theorem which allows limits to be found if a function is squeezed between two others with known limits. Examples are provided to illustrate finding limits through algebraic manipulation and using these theorems.
Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard ...Marco Benini
A novel approach to giving an interpretation of logic inside category theory. This work has been developed as part of my sabbatical Marie Curie fellowship in Leeds.
Presented at the Logic Seminar, School of Mathematics, University of Leeds (2012).
The document discusses multi-variable functions and how to visualize them. It defines Rn as the set of n-tuples of real numbers. A n-variable real valued function maps from Rn to the real numbers. Level curves are used to visualize functions of two variables by setting z equal to constants and plotting the resulting equations in the x-y plane. This produces families of curves that depict the shape of the surface. Examples show how different functions produce different level curve shapes like ellipses, hyperbolas or parabolas.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
1) The document discusses formulas for calculating area, volume, fluid pressure, and work based on the cross-sectional lengths and areas of regions and solids.
2) It provides examples of calculating the area of regions bounded by functions, the volume of solids of revolution, fluid pressure on a plate, and work needed to pump water.
3) The key concepts are using integrals to calculate quantities by summing cross-sectional lengths or areas and defining these lengths and areas based on the geometry of regions and solids.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
This document discusses calculating volumes of solids formed by rotating line segments in space. It provides formulas for finding horizontal and vertical distances between points on the real line and in the xy-plane. These distances are used to calculate areas of surfaces formed when line segments are revolved. Examples are provided to demonstrate expressing variables as functions of each other and finding horizontal and vertical distances between points on graphs of equations.
The document describes Lagrange multipliers, which are used to find the extrema (maximum and minimum points) of a function subject to a constraint. Specifically:
1) A function z=f(x,y) defines a surface, and an equation g(x,y)=0 defines a curve on the xy-plane.
2) The points where this curve intersects the surface form a "trail".
3) The extrema on this trail occur where the gradients of the surface and constraint are parallel (or equivalently where their normals are parallel), allowing the use of Lagrange multipliers to solve the constrained optimization problem.
This document defines categories, functors, and natural transformations in category theory. It begins by discussing the "size problem" in naively defining categories and introduces the concept of a universe to address this. Categories are then defined as classes of objects and sets of arrows between objects, satisfying composition and identity laws. Functors map categories to categories by mapping objects and arrows, preserving structure. Natural transformations relate functors by assigning morphisms between their actions on objects. The Yoneda lemma and Godement products of natural transformations are also introduced.
This document provides an overview of key calculus concepts and formulas taught in a Calculus I course at Miami Dade College - Hialeah Campus. The topics covered include limits and derivatives, integration, optimization techniques, and applications of calculus to economics, business, physics, and other fields. The document is intended as a study guide for students in the Calculus I class taught by Professor Mohammad Shakil.
The document discusses partial derivatives. It defines a partial derivative as the slope of a curve intersecting a surface at a point, where the curve is obtained by fixing one of the variables in the surface equation. The partial derivative with respect to x is the slope of the curve intersecting when y is fixed, and vice versa for the partial derivative with respect to y. Examples are provided to demonstrate calculating partial derivatives algebraically and finding equations of tangent lines using partial derivatives.
The document discusses graphs in polar coordinates. It begins by reviewing how polar coordinates (r, θ) represent a point P, where r is the distance from the origin and θ is the angle relative to the x-axis. It then discusses how to convert between rectangular (x, y) and polar (r, θ) coordinates. The document explains that polar equations relate r and θ, giving the example of r = θ which graphs as an Archimedean spiral. It also discusses the graphs of constant equations like r = c, which is a circle, and θ = c, which is a line. The document concludes by explaining how to graph other polar equations by plotting points.
The document discusses 3D coordinate systems. It explains that a z-axis is added perpendicular to the x- and y-axes to form a 3D coordinate system. There are two ways to orient the z-axis, known as the right-hand system and left-hand system. Every point in 3D space can be located using an ordered triple (x, y, z). The document also discusses the three coordinate planes and provides an example of sketching the graph of an equation in 3D space.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
Here are the problems from the slides with their solutions:
1. Find the slope of the line tangent to the graph of the function f(x) = x^2 - 5x + 8 at the point P(1,4).
Slope = -3
2. Find the equation of the tangent line to the curve f(x) = 2x^2 - 3 at the point P(1,-1) using point-slope form.
y - (-1) = 4(x - 1)
3. Find the equation of the tangent line to the curve f(x) = x + 6 at the point P(3,3) using point-slope form.
y
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
8.further calculus Further Mathematics Zimbabwe Zimsec Cambridgealproelearning
This document discusses techniques for approximating integrals, including the trapezium rule and Simpson's rule. The trapezium rule approximates the area under a curve as the sum of trapezoidal areas formed by the function values at the endpoints of subintervals. Simpson's rule approximates the area as the sum of triangular areas, weighted differently for even and odd terms, formed by the function values at three evenly spaced points in each subinterval. Examples are given to demonstrate applying these rules to approximate definite integrals. The Simpson's rule is generally more accurate because it approximates the curve by a quadratic rather than a straight line as in the trapezium rule.
19 more parabolas a& hyperbolas (optional) xmath260
After dividing the general quadratic equation Ax2 + By2 + Cx + Dy = E by A, three types of conic sections can be obtained:
1) Parabolas occur when B = 0, resulting in equations of the form 1x2 + #x + #y = #.
2) Circles occur when A = B, resulting in the equation 1x2 + 1y2 = 1.
3) Hyperbolas occur when A and B have opposite signs, resulting in equations of the form 1x2 + ry2 + #x + #y = # with r < 0. Hyperbolas have two foci and asymptotes, and points on the hyperbola have
This document discusses finding the maximum and minimum values of functions, as well as points of inflection. It provides instructions on how to find the coordinates of turning points by setting the derivative of the function equal to zero and solving. It also discusses how to determine if a turning point is a maximum or minimum by taking the second derivative and checking if it is positive or negative. The document concludes by giving examples of how to apply these concepts to optimization word problems involving areas, volumes, or other quantities that need to be maximized or minimized under certain constraints.
The document discusses quadratic functions and how to graph them. It defines a quadratic function as a function of the form f(x) = ax^2 + bx + c, where a ≠ 0. It explains that the graph of such a function is a parabola, and provides steps to find characteristics of the parabola like the vertex, axis of symmetry, x-intercepts, and y-intercept in order to graph the function. Examples are included to demonstrate applying these steps to graph specific quadratic functions.
This document defines continuity and uniform continuity of functions. A function f is continuous on a set S if small changes in the input x result in small changes in the output f(x). A function is uniformly continuous if the same relationship holds for all inputs and outputs simultaneously, not just for a fixed input. Several examples are provided to illustrate the difference. The key difference is that a continuous function may depend on the specific input point, while a uniformly continuous function does not. Functions that satisfy a Lipschitz inequality are proven to be uniformly continuous.
The document discusses the properties of parabolas that open up or down and parabolas that open left or right. It provides the general form and standard form of each, noting that a positive value of a in the standard form indicates the direction the parabola opens (up if discussing vertical parabolas, right if discussing horizontal parabolas) while a negative a indicates the opposite direction. It also identifies the axis of symmetry formula for each. Examples are given to demonstrate finding the direction a parabola opens, its vertex, and axis of symmetry.
This document appears to be a collection of prompts related to calculus concepts and problems. It includes prompts about finding derivatives, integrals, limits, solving differential equations, finding areas under curves, rates of change, and other common calculus topics. The "You think..." sections after each prompt seem to suggest possible thoughts or steps one may have when encountering those types of problems.
The document summarizes the Fundamental Theorem of Calculus, which establishes a connection between computing integrals (areas under curves) and computing derivatives. It shows that if f is continuous on an interval [a,b] and F is defined by integrating f, then F' = f. Graphs and examples are provided to justify this theorem geometrically and demonstrate its applications to computing derivatives and integrals.
1) The document discusses formulas for calculating area, volume, fluid pressure, and work based on the cross-sectional lengths and areas of regions and solids.
2) It provides examples of calculating the area of regions bounded by functions, the volume of solids of revolution, fluid pressure on a plate, and work needed to pump water.
3) The key concepts are using integrals to calculate quantities by summing cross-sectional lengths or areas and defining these lengths and areas based on the geometry of regions and solids.
Lesson 2: A Catalog of Essential Functions (slides)Matthew Leingang
This document provides an overview of different types of functions including: linear, polynomial, rational, power, trigonometric, and exponential functions. It discusses representing functions verbally, numerically, visually, and symbolically. Key topics covered include transformations of functions through shifting graphs vertically and horizontally, as well as composing multiple functions.
This document discusses calculating volumes of solids formed by rotating line segments in space. It provides formulas for finding horizontal and vertical distances between points on the real line and in the xy-plane. These distances are used to calculate areas of surfaces formed when line segments are revolved. Examples are provided to demonstrate expressing variables as functions of each other and finding horizontal and vertical distances between points on graphs of equations.
The document describes Lagrange multipliers, which are used to find the extrema (maximum and minimum points) of a function subject to a constraint. Specifically:
1) A function z=f(x,y) defines a surface, and an equation g(x,y)=0 defines a curve on the xy-plane.
2) The points where this curve intersects the surface form a "trail".
3) The extrema on this trail occur where the gradients of the surface and constraint are parallel (or equivalently where their normals are parallel), allowing the use of Lagrange multipliers to solve the constrained optimization problem.
This document defines categories, functors, and natural transformations in category theory. It begins by discussing the "size problem" in naively defining categories and introduces the concept of a universe to address this. Categories are then defined as classes of objects and sets of arrows between objects, satisfying composition and identity laws. Functors map categories to categories by mapping objects and arrows, preserving structure. Natural transformations relate functors by assigning morphisms between their actions on objects. The Yoneda lemma and Godement products of natural transformations are also introduced.
This document provides an overview of key calculus concepts and formulas taught in a Calculus I course at Miami Dade College - Hialeah Campus. The topics covered include limits and derivatives, integration, optimization techniques, and applications of calculus to economics, business, physics, and other fields. The document is intended as a study guide for students in the Calculus I class taught by Professor Mohammad Shakil.
The document discusses partial derivatives. It defines a partial derivative as the slope of a curve intersecting a surface at a point, where the curve is obtained by fixing one of the variables in the surface equation. The partial derivative with respect to x is the slope of the curve intersecting when y is fixed, and vice versa for the partial derivative with respect to y. Examples are provided to demonstrate calculating partial derivatives algebraically and finding equations of tangent lines using partial derivatives.
The document discusses graphs in polar coordinates. It begins by reviewing how polar coordinates (r, θ) represent a point P, where r is the distance from the origin and θ is the angle relative to the x-axis. It then discusses how to convert between rectangular (x, y) and polar (r, θ) coordinates. The document explains that polar equations relate r and θ, giving the example of r = θ which graphs as an Archimedean spiral. It also discusses the graphs of constant equations like r = c, which is a circle, and θ = c, which is a line. The document concludes by explaining how to graph other polar equations by plotting points.
The document discusses 3D coordinate systems. It explains that a z-axis is added perpendicular to the x- and y-axes to form a 3D coordinate system. There are two ways to orient the z-axis, known as the right-hand system and left-hand system. Every point in 3D space can be located using an ordered triple (x, y, z). The document also discusses the three coordinate planes and provides an example of sketching the graph of an equation in 3D space.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
Here are the problems from the slides with their solutions:
1. Find the slope of the line tangent to the graph of the function f(x) = x^2 - 5x + 8 at the point P(1,4).
Slope = -3
2. Find the equation of the tangent line to the curve f(x) = 2x^2 - 3 at the point P(1,-1) using point-slope form.
y - (-1) = 4(x - 1)
3. Find the equation of the tangent line to the curve f(x) = x + 6 at the point P(3,3) using point-slope form.
y
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
8.further calculus Further Mathematics Zimbabwe Zimsec Cambridgealproelearning
This document discusses techniques for approximating integrals, including the trapezium rule and Simpson's rule. The trapezium rule approximates the area under a curve as the sum of trapezoidal areas formed by the function values at the endpoints of subintervals. Simpson's rule approximates the area as the sum of triangular areas, weighted differently for even and odd terms, formed by the function values at three evenly spaced points in each subinterval. Examples are given to demonstrate applying these rules to approximate definite integrals. The Simpson's rule is generally more accurate because it approximates the curve by a quadratic rather than a straight line as in the trapezium rule.
19 more parabolas a& hyperbolas (optional) xmath260
After dividing the general quadratic equation Ax2 + By2 + Cx + Dy = E by A, three types of conic sections can be obtained:
1) Parabolas occur when B = 0, resulting in equations of the form 1x2 + #x + #y = #.
2) Circles occur when A = B, resulting in the equation 1x2 + 1y2 = 1.
3) Hyperbolas occur when A and B have opposite signs, resulting in equations of the form 1x2 + ry2 + #x + #y = # with r < 0. Hyperbolas have two foci and asymptotes, and points on the hyperbola have
This document discusses finding the maximum and minimum values of functions, as well as points of inflection. It provides instructions on how to find the coordinates of turning points by setting the derivative of the function equal to zero and solving. It also discusses how to determine if a turning point is a maximum or minimum by taking the second derivative and checking if it is positive or negative. The document concludes by giving examples of how to apply these concepts to optimization word problems involving areas, volumes, or other quantities that need to be maximized or minimized under certain constraints.
The document discusses quadratic functions and how to graph them. It defines a quadratic function as a function of the form f(x) = ax^2 + bx + c, where a ≠ 0. It explains that the graph of such a function is a parabola, and provides steps to find characteristics of the parabola like the vertex, axis of symmetry, x-intercepts, and y-intercept in order to graph the function. Examples are included to demonstrate applying these steps to graph specific quadratic functions.
This document defines continuity and uniform continuity of functions. A function f is continuous on a set S if small changes in the input x result in small changes in the output f(x). A function is uniformly continuous if the same relationship holds for all inputs and outputs simultaneously, not just for a fixed input. Several examples are provided to illustrate the difference. The key difference is that a continuous function may depend on the specific input point, while a uniformly continuous function does not. Functions that satisfy a Lipschitz inequality are proven to be uniformly continuous.
The document discusses the properties of parabolas that open up or down and parabolas that open left or right. It provides the general form and standard form of each, noting that a positive value of a in the standard form indicates the direction the parabola opens (up if discussing vertical parabolas, right if discussing horizontal parabolas) while a negative a indicates the opposite direction. It also identifies the axis of symmetry formula for each. Examples are given to demonstrate finding the direction a parabola opens, its vertex, and axis of symmetry.
This document appears to be a collection of prompts related to calculus concepts and problems. It includes prompts about finding derivatives, integrals, limits, solving differential equations, finding areas under curves, rates of change, and other common calculus topics. The "You think..." sections after each prompt seem to suggest possible thoughts or steps one may have when encountering those types of problems.
The document summarizes the Fundamental Theorem of Calculus, which establishes a connection between computing integrals (areas under curves) and computing derivatives. It shows that if f is continuous on an interval [a,b] and F is defined by integrating f, then F' = f. Graphs and examples are provided to justify this theorem geometrically and demonstrate its applications to computing derivatives and integrals.
The document defines the derivative of a function and discusses:
- The definition of the derivative as the limit of the slope between two points as they approach each other.
- Notation used to represent derivatives, including f'(x), dy/dx, and df/dx.
- How the graph of a function's derivative f' relates to the graph of the original function f - where f' is positive/negative/zero corresponds to parts of f that are increasing/decreasing/at an extremum.
- How to graph f given a graph of its derivative f' by sketching the curve that matches the behavior of f' at each point.
- One-sided derivatives at endpoints of functions defined
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand the derivative of constants, the constant multiple rule, the sum rule, the difference rule, and derivatives of sine and cosine. Examples are provided to find the derivatives of squaring, cubing, square root, and cube root functions using the definition of the derivative. Graphs and properties of these functions and their derivatives are also discussed.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
This document contains lecture notes on optimization problems from a Calculus I class. It provides announcements about upcoming exams and room changes, then outlines examples of optimization problems involving addition with constraints, finding distances, and finding maximal areas of rectangles inscribed in triangles. It reviews methods for finding extrema like the closed interval method, first derivative test, and second derivative test. It then works through the examples in detail, finding critical points and classifying them to determine the optimal solutions.
The basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
This document contains notes from a calculus class lecture on optimization problems. It provides examples of optimization problems involving addition with a constraint, finding distance, and finding the maximum area rectangle that can be inscribed in a 3-4-5 right triangle. It also reviews concepts like the closed interval method, first derivative test, and second derivative test for finding extrema of functions.
This document contains lecture notes on the Fundamental Theorem of Calculus. It begins with announcements about the final exam date and current grade distribution. The outline then reviews the Evaluation Theorem and introduces the First and Second Fundamental Theorems of Calculus. It provides examples of how the integral can represent total change in concepts like distance, cost, and mass. Biographies are also included of mathematicians like Gregory, Barrow, Newton, and Leibniz who contributed to the development of calculus.
This document contains notes from a calculus class. It provides the outline and key points about the Fundamental Theorem of Calculus. It discusses the first and second Fundamental Theorems of Calculus, including proofs and examples. It also provides brief biographies of several important mathematicians that contributed to the development of calculus, including the Fundamental Theorem of Calculus, such as Isaac Newton, Gottfried Leibniz, James Gregory, and Isaac Barrow.
The document discusses key concepts related to limits, continuity, and differentiation. It defines what it means for a variable x to approach a finite number a or infinity, and provides the formal definitions of one-sided limits and two-sided limits. It also discusses indeterminate forms when limits take on forms like 0/0, infinity/infinity, or infinity - infinity. The document outlines several properties of limits, including limits of even and odd functions. It distinguishes between the limit of a function as x approaches a, denoted limx→af(x), versus the function value at that point, f(a). Finally, it states standard theorems about limits, such as the sum and product of two functions whose limits exist
This document provides an overview of key calculus concepts including:
- Functions and function notation which are fundamental to calculus
- Limits which allow defining new points from sequences and are essential to calculus concepts like derivatives and integrals
- Derivatives which measure how one quantity changes in response to changes in another related quantity
- Types of infinity and limits involving infinite quantities or areas
The document defines functions, limits, derivatives, and infinity, and provides examples to illustrate these core calculus topics. It lays the groundwork for further calculus concepts to be covered like integrals, derivatives of more complex functions, and applications of limits, derivatives, and infinity.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
The document provides an overview of Calculus I taught by Professor Matthew Leingang at New York University. It outlines key topics that will be covered in the course, including different classes of functions, transformations of functions, and compositions of functions. The first assignments are due on January 31 and February 2, with first recitations on February 3. The document uses examples to illustrate concepts like linear functions, other polynomial functions, and trigonometric functions. It also explains how vertical and horizontal shifts can transform the graph of a function.
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms for those who already suffer from conditions like anxiety and depression.
This document contains 5 problems involving finding terms in expansions of polynomials. The problems involve finding specific terms that contain a given power of x in expansions of polynomials such as (3x^4 - 1)^9, (-x^3 + 2)^6, (x + 1)^3x, (x + 1)^x, and determining the value of m if one term in the expansion of (2x - m)^7 is -262500x^2y^5.
The document discusses binomial expansion, which is the process of multiplying out terms with two variables according to their power using the binomial theorem. It provides examples of expanding binomial expressions like (x + y)2, (x + y)3, and (x + y)4. It also notes that the sum of the exponents in each term equals the overall power, and the number of terms is always one more than the power. Finally, it provides the binomial theorem for expanding any binomial expression and finding a particular term.
This document discusses how to calculate arrangements when some items must be together or apart. It explains that when items need to be together, they should be counted as a single item to reduce the total items being arranged. Then the total number of arrangements is calculated by finding the total possible arrangements and subtracting the arrangements that do not satisfy the constraints of certain items being together or apart. Examples provided include arranging people in a row when some must or cannot sit together and arranging books on a shelf keeping books of each subject together.
Permutations refer to arrangements of objects in a definite order. Some key points:
- Permutations are represented by "nPn" where n is the total number of objects and r is the number being arranged.
- Permutations are used to calculate possibilities like license plates, phone numbers, and locker combinations.
- Restrictions like starting/ending conditions or requiring alternating arrangements reduce the number of possible permutations.
- Objects that are identical only count once toward the total number of permutations rather than being distinguishable.
Factorial notation represents the product of all positive integers less than or equal to the given number. For example, 5! = 5 x 4 x 3 x 2 x 1 and 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. The document also provides examples of simplifying factorials without a calculator by using properties such as 5! + 4! = 6 x 4! and (k + 1)! + k! = (k + 2)k!.
This document provides examples and explanations of the fundamental counting principle and addition counting principle to solve combinatorics problems. It gives 8 examples of using the fundamental counting principle to count the number of possible outcomes of independent events. These include counting the number of volleyball shoe combinations, outfits that can be created from different clothing items, ways to select committees from groups of people, and 3-digit numbers with no repeating digits. It also provides 5 examples of using the addition counting principle to count outcomes when events are dependent, such as selecting a president and vice president of opposite sexes from a group.
This document discusses graphing composite functions. It provides examples of determining the composite functions f(g(x)) and g(f(x)) for various functions f(x) and g(x), sketching the graphs of the composite functions, and stating their domains. It also gives examples of determining possible functions f(x) and g(x) that satisfy given composite functions.
1. The document discusses composite functions, which involve combining two functions f(x) and g(x) where the output of one is used as the input of the other. It provides examples of evaluating composite functions using tables and graphs.
2. Key steps for evaluating composite functions are: 1) Substitute the inner function into the outer function and 2) Simplify the expression. Order matters as f(g(x)) and g(f(x)) may have different values.
3. Examples are worked through to find composite functions given basic functions like f(x) = x + 1 and g(x) = 2x as well as more complex rational functions.
The graph is a linear function with a domain of all real numbers and a range of real numbers greater than or equal to 3. The graph is a line with a y-intercept of 3 that increases at a rate of 1 as x increases.
Rational functions are functions of the form f(x) = p(x)/q(x) where p(x) and q(x) are polynomials. For example, comparing rational functions like 2x/(x^2 - 4) and (x-1)/(x+1). Horizontal asymptotes of rational functions occur when the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
This document discusses combining functions by graphing. When two functions f(x) and g(x) are combined, their graphs are overlayed on the same coordinate plane. The result is a new combined function where the output is determined by applying both functions f(x) and g(x) to the same input x.
This document discusses how to find the sum, difference, product, and quotient of functions. The sum of functions is found by adding the y-coordinates of each function. The difference is found by subtracting the y-coordinates. The product is represented as h(x) = f(x)g(x) and the quotient is represented as h(x) = f(x)/g(x). Examples are provided for adding and subtracting functions.
The document outlines a mental math test covering polynomials. It includes short answer questions testing long division, synthetic division, the remainder theorem, and finding the degree, leading coefficient, and y-intercept of polynomials. The test also covers matching graphs to polynomial equations and word problems involving fully factoring polynomials and two graphs. Multiple choice questions will require explaining solutions, while long answer questions involve fully factoring polynomials and word problems.
The document contains two polynomial word problems. The first asks to write a function V(x) to express the volume of a box with dimensions x, x+2, x+10 in terms of x, and find possible x values if the volume is 96 cm^3. The second problem describes a block of ice that is initially 3 ft by 4 ft by 5 ft, and asks to write a function to model reducing each dimension by the same amount to reach a volume of 24 ft^3, and determine how much to remove from each dimension.
The document provides 3 polynomial word problems: 1) finding the equation for a polynomial given its graph f(x) = -(x - 2)2(x + 1), 2) determining the polynomial P(x) when divided by (x - 3) with a quotient of 2x^2 + x - 6 and remainder of 4, and 3) finding the value of a if (x - 2) is a factor of ax^3 + 4x^2 + x - 2. It also gives a 4th problem of determining the value of k when 2x^3 + kx^2 - 3x + 2 is divided by x - 2 with a remainder of 4.
Polynomial functions are described by their degree and have certain characteristics. The graph of a polynomial is smooth and continuous without sharp corners. Odd degree polynomials rise on the left and fall on the right, while even degree polynomials rise on both sides. The number of x-intercepts and local maxima/minima are limited by the degree. Polynomials can be matched based on their degree, leading coefficient, even/odd nature, and number of x-intercepts and local extrema. The x-intercepts of a polynomial correspond to the roots of the equation, and a repeated root indicates a zero of higher multiplicity which affects the graph.
13. When you see…
Find the interval where the
slope of f (x) is increasing
You think…
14. Slope of f (x) is increasing
Find the derivative of f ’(x) = f “ (x)
Set numerator and denominator = 0
to find critical points
Make sign chart of f “ (x)
Determine where it is positive
22. Find inflection points
Express f “ (x) as a fraction
Set numerator and denominator = 0
Make a sign chart of f “ (x)
Find where it changes sign
( + to - ) or ( - to + )
37. When you see…
Find the absolute
minimum of f(x) on [a, b]
You think…
38. Find the absolute minimum of f(x)
a) Make a sign chart of f ’(x)
b) Find all relative maxima
c) Plug those values into f (x)
d) Find f (a) and f (b)
e) Choose the largest of c) and d)
39. When you see…
Show that a piecewise
function is differentiable
at the point a where the
function rule splits
You think…
40. Show a piecewise function is
differentiable at x=a
First, be sure that the function is continuous at
x =a .
Take the derivative of each piece and show that
lim− f ′( x ) = lim f ′( x )
x →a x →a +
43. When you see…
Given v(t), find how far a
particle travels on [a, b]
You think…
44. Given v(t), find how far a particle
travels on [a,b]
b
v ()
∫t dt
Find a
45. When you see…
Find the average
velocity of a particle
on [a, b]
You think…
46. Find the average rate of change on
[a,b]
b
∫) s(s(
v(
t dt
b) )
−a
Find a
b−
a
=
b−
a
47. When you see…
Given v(t), determine if a
particle is speeding up at
t=a
You think…
48. Given v(t), determine if the particle is
speeding up at t=a
Find v (k) and a (k).
Multiply their signs.
If positive, the particle is speeding up.
If negative, the particle is slowing down
50. Given v(t) and s(0), find s(t)
s()=
t ∫ v ()dt + C
t
Plug in t = 0 to find C
51. When you see…
Show that Rolle’s
Theorem holds on [a, b]
You think…
52. Show that Rolle’s Theorem holds on
[a,b]
Show that f is continuous and differentiable
on the interval
If f () f ( , then find some c in [b]
a = b) a,
such that f ′c)0.
(=
53. When you see…
Show that the Mean
Value Theorem holds
on [a, b]
You think…
54. Show that the MVT holds on [a,b]
Show that f is continuous and differentiable
on the interval.
Then find some c such that
f (−(
b) f a )
f ′c)
(= b − .a
56. Find the domain of f(x)
Assume domain is (−∞, ∞ ).
Domain restrictions: non-zero denominators,
Square root of non negative numbers,
Log or ln of positive numbers
62. Find f ‘( x) by definition
f ( +h ) ()
x −f x
() h→
f ′ x =lim
0 h
or
f () ()
x −f a
() x → x −a
f ′ x =lim
a
63. When you see…
Find the derivative of
the inverse of f(x) at x = a
You think…
64. Derivative of the inverse of f(x) at x=a
Interchange x with y.
dy
Solve for dx implicitly (in terms of y).
Plug your x value into the inverse relation
and solve for y.
dy
Finally, plug that y into your dx .
65. When you see…
y is increasing
proportionally to y
You think…
66. . y is increasing proportionally to y
dy
=
ky
dt
translating to
y=
Ce kt
67. When you see…
Find the line x = c that
divides the area under
f(x) on [a, b] into two
equal areas
You think…
68. Find the x=c so the area under f(x) is
divided equally
c b
∫ f (x )dx =∫ f (x )dx
a c
75. When you see…
The line y = mx + b is
tangent to f(x) at (a, b)
You think…
76. y = mx+b is tangent to f(x) at (a,b)
.
Two relationships are true.
The two functions share the same
slope ( m = f ′( x ) )
and share the same y value at x1 .
80. Area using right Riemann sums
A = base[ x1 + x 2 + x3 + ... + x n ]
81. When you see…
Find area using
midpoint rectangles
You think…
82. Area using midpoint rectangles
Typically done with a table of values.
Be sure to use only values that are
given.
If you are given 6 sets of points, you can
only do 3 midpoint rectangles.
84. Area using trapezoids
base
A= [ x0 + 2 x1 + 2 x 2 + ... + 2 x n − 1 + x n ]
2
This formula only works when the base is the
same.
If not, you have to do individual trapezoids
88. Meaning of the integral of f(t) from a to x
The accumulation function –
accumulated area under the
function f ( x )
starting at some constant a
and ending at x
89. When you see…
Given a base, cross
sections perpendicular to
the x-axis that are
squares
You think…
90. Semi-circular cross sections
perpendicular to the x-axis
The area between the curves typically
is the base of your square.
b
So the volume is ∫ (base )dx
2
a
91. When you see…
Find where the tangent
line to f(x) is horizontal
You think…
98. Approximate f(0.1) using tangent line
to f(x) at x = 0
Find the equation of the tangent line to f
using y − y1 = m( x − x1 )
where m = f ′( 0) and the point is ( 0, f ( 0) ) .
Then plug in 0.1 into this line.
Be sure to use an approximation ( ≈) sign.
99. When you see…
Given the value of F(a)
and the fact that the
anti-derivative of f is F,
find F(b)
You think…
100. Given F(a) and the that the
anti-derivative of f is F, find F(b)
Usually, this problem contains an antiderivative
you cannot take. Utilize the fact that if F (x )
is the antiderivative of f,
b
then ∫F (x )dx =F (b) −F (a ) .
a
Solve for F (b ) using the calculator
to find the definite integral
103. When you see…
b b
Given ∫ f (x )dx , find ∫ [f (x )+ k ]dx
a a
You think…
104. Given area under a curve and vertical
shift, find the new area under the curve
b b b
∫ [ f ( x ) + k ] dx = ∫ f ( x )dx + ∫ kdx
a a a
105. When you see…
Given a graph of f '( x )
find where f(x) is
increasing
You think…
106. Given a graph of f ‘(x) , find where f(x) is
increasing
Make a sign chart of f ′( x )
Determine where f ′( x ) is positive
107. When you see…
Given v(t) and s(0), find the
greatest distance from the
origin of a particle on [a, b]
You think…
108. Given v(t) and s(0), find the greatest distance from
the origin of a particle on [a, b]
Generate a sign chart of v( t ) to find
turning points.
Integrate v( t ) using s ( 0 ) to find the
constant to find s( t ) .
Find s(all turning points) which will give
you the distance from your starting point.
Adjust for the origin.
109. When you see…
Given a water tank with g gallons
initially being filled at the rate of
F(t) gallons/min and emptied at
the rate of E(t) gallons/min on
[t1 , t2 ] , find
110. a) the amount of water in
the tank think…
You at m minutes
111. Amount of water in the tank at t minutes
t2
g + ∫( F (t ) − E ( t ) )dt
t
112. b) the rate the water
amount is changing
at m
You think…
113. Rate the amount of water is
changing at t = m
m
d
∫ ( F ( t ) − E ( t ) )dt = F ( m ) − E ( m )
dt t
114. c) the time when the
water is at a minimum
You think…
115. The time when the water is at a minimum
F ( m ) − E ( m ) = 0,
testing the endpoints as well.
116. When you see…
Given a chart of x and f(x)
on selected values between
a and b, estimate f '( x ) where
c is between a and b.
You think…
117. Straddle c, using a value k greater
than c and a value h less than c.
f (k ) −f (h )
so f ′c) ≈
(
k− h
118. When you see…
dy
Given dx , draw a
slope field
You think…
119. Draw a slope field of dy/dx
Use the given points
dy
Plug them into dx ,
drawing little lines with the
indicated slopes at the points.
120. When you see…
Find the area between
curves f(x) and g(x) on
[a,b]
You think…
121. Area between f(x) and g(x) on [a,b]
b
A = ∫[ f ( x ) − g ( x )]dx
a
,
assuming f (x) > g(x)
122. When you see…
Find the volume if the
area between the curves
f(x) and g(x) is
rotated about the x-axis
You think…
123. Volume generated by rotating area between
f(x) and g(x) about the x-axis
∫ [( f ( x ) ) ]
b
− ( g ( x ) ) dx
2 2
A=
a
assuming f (x) > g(x).