2.6 Related Rates
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This document defines and provides examples of related rate problems. Related rate problems involve calculating the rates of change of two or more related variables with respect to time. The document outlines the steps to solve related rate problems: 1) identify changing quantities, 2) write an equation relating the quantities, 3) differentiate both sides with respect to time, 4) solve for the required quantity and substitute known values. Examples include calculating the rate of change of a dog's circular puddle area, a melting snowball's radius, the top of a slipping ladder, and the height of a conical sand pile.
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So in summary, given the measurements A, a and c where a < c, there will be 0 triangles formed if a < h, 1 triangle if a = h, and 2 triangles if a > h.
6.2 law of cosines
This document provides instruction on using the Law of Cosines to solve triangles. It begins by explaining that the Law of Sines cannot be used to solve triangles when the side-side-side (SSS) or side-angle-side (SAS) configurations are given. The Law of Cosines formulas are then introduced. Examples are worked through demonstrating how to use the Law of Cosines to solve triangles in SSS and SAS formations. Key steps like verifying a triangle can be formed from given SSS measurements and properly applying order of operations are emphasized. Students are assigned homework problems applying these concepts.
Second Derivative Information
The second derivative of a function f(x) provides information about the concavity and points of inflection of f(x). It indicates the intervals where f(x) is concave up or down. A curve is concave up where the second derivative is positive and concave down where the second derivative is negative. Points of inflection are where the concavity changes from up to down or vice versa, and occur where the second derivative is zero or undefined.
Definite Integral Review
This document provides an overview of key concepts in calculus including the definite integral, properties of definite integrals, the Mean Value Theorem for Integrals, the Average Value Theorem, the Fundamental Theorem of Calculus parts 1 and 2, and the Trapezoidal Rule for approximating definite integrals. It defines integrals, discusses how to evaluate them on a TI-84 calculator, and lists properties such as additivity, constant multiples, and order of integration. It also introduces concepts like finding the average value of a function over an interval using the integral, and relating the derivative of an integral to the original function.
4.4 Fundamental Theorem of Calculus
This document discusses the mean value theorem and how it can be used to find the average value of a function over an interval. It states that for a continuous function, there exists some value c within the interval where the area under the curve between the bounds is equal to the value of the function at c multiplied by the width of the interval. It then works through an example of finding the average value of the function f(x)=3x^2-2x over the interval [1,4].
4.3 The Definite Integral
The document discusses notation and properties for definite integrals. It defines the definite integral from a to b of f(x) dx as the area under the curve of f(x) between the x-axis and the limits of a and b. It lists five properties of definite integrals: 1) the order of integration does not matter, 2) the integral from a to a of any function f(x) is equal to zero, 3) a constant can be pulled out of the integral, 4) integrals can be added or subtracted, and 5) a definite integral over an interval can be broken into a sum of integrals over subintervals.
Lesson 3.3 First Derivative Information
The first derivative of a function f(x) indicates where the function is increasing, decreasing, or constant. If f'(x) is positive, f(x) is increasing, and if f'(x) is negative, f(x) is decreasing. If f'(x) is 0, f(x) is constant. To determine where a function is increasing or decreasing, identify the critical numbers and determine the sign of f'(x) in intervals between them. The first derivative test also allows identification of relative extrema - if f'(x) changes from negative to positive at a critical number c, then f(c) is a relative minimum, and if it changes from positive to negative, f(c
Lesson 3.2 Rolle and Mean Value Theorems
Rolle's Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), with f(a) = f(b), then there exists at least one value c in the open interval where the derivative f'(c) is equal to 0. The Mean Value Theorem similarly guarantees the existence of at least one value c where the derivative is equal to the average rate of change of the function over the interval, (f(b)-f(a))/(b-a).
Lesson 3.3 3.4 - 1 st and 2nd Derivative Information
1) The document discusses tests for determining whether a function is increasing, decreasing, or constant based on the sign of the first derivative on an interval.
2) It also describes how to use the first and second derivative tests to determine if a critical point is a relative maximum or minimum. The tests analyze how the sign of the first or second derivative changes at the critical point.
3) Finally, the document outlines the test for concavity and points of inflection based on the sign of the second derivative on an interval. A change in the sign of the second derivative indicates an inflection point.
3.5 3.6 exp-log models 13-14
1. This document discusses several common mathematical models involving exponential and logarithmic functions, including exponential growth/decay models, logistic growth models, Gaussian models, logarithmic models, and Newton's Law of Cooling.
2. Examples are provided to illustrate how to set up and solve problems using each type of model. For instance, an example uses an exponential growth model to predict when the world's population will reach 7.1 billion people.
3. Guidance is also given on using graphing utilities and calculators to analyze exponential and logarithmic relationships. Instructions for performing exponential regression on a TI-84 calculator are provided.
3.1 Extreme Values of Functions
This document defines and discusses absolute and local extreme values of functions. It states that absolute extrema (global maximum and minimum values) can occur at endpoints or interior points of an interval, but a function is not guaranteed to have an absolute max or min on every interval. The Extreme Value Theorem says that if a function is continuous on a closed interval, it will have both an absolute maximum and minimum value. Local extrema are defined as maximum or minimum values within an open neighborhood of a point, and the theorem is presented that if a function has a local extremum at an interior point where the derivative exists, the derivative must be zero at that point. Critical points are defined as points where the derivative is zero or undefined. Methods
3.5 EXP-LOG MODELS
The document discusses various mathematical models involving exponential and logarithmic functions, including exponential growth and decay models, logistic growth models, Gaussian models, and logarithmic models. It provides examples of how each type of model can be used, such as modeling world population growth with an exponential function, modeling the spread of a virus on a college campus with a logistic function, and modeling cooling of a heated object using Newton's Law of Cooling. The document also discusses how to graph and solve problems involving these different mathematical models.
Larson 4.4
This document discusses evaluating trigonometric functions of any angle by using reference angles. It defines trigonometric functions for any real value of the angle θ. For angles greater than 90° or less than 0°, the values can be determined from the acute reference angle. Examples show how to find the reference angle and then use it to evaluate trigonometric functions in different quadrants, including for nonacute angles.
Larson 4.3
This document discusses trigonometric functions and their applications. It defines the six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - using ratios of sides of a right triangle. Examples are provided to evaluate the trig functions of given angles and to use identities to relate functions. The document also discusses applications of solving right triangles by using trig functions when given angle and side length information.
Larson 4.2
The unit circle relates real numbers to points on a circle of radius 1 centered at the origin. Each real number t corresponds to a point (x, y) on the circle, allowing the definition of the six trigonometric functions in terms of x and y. The trigonometric functions sine and cosine are periodic with period 2π, meaning their values repeat every 2π units. Their domains are all real numbers and their ranges are between -1 and 1.
Larson 4.1
The document describes angles and their measurement in radians and degrees. It defines an angle, measures angles in radians using central angles of a circle, and defines the relationship between radians and degrees. Examples show converting between radians and degrees, finding coterminal angles, complementary and supplementary angles, and calculating arc length and linear speed using angular measurements.
Derivative rules ch2
This document provides the basic derivative rules from Chapter 2. It lists various expressions and their derivatives on subsequent slides, including the derivatives of sums, products, quotients, composite functions, powers, trigonometric functions like sine, cosine, tangent, cotangent, secant and cosecant, and constants. The derivatives covered are: du/dv, df(g(x))/dx, dn/dx, d/dx cot(u), d/dx cu, d/dx tan(u), d/dx (u*v), d/dx c, d/dx cos(u), d/dx sec(u), dx/dx, and d/dx csc(
1.4 Continuity
The document discusses continuity of functions at a point c. It defines continuity as a function being defined at c, the limit existing at c, and the function value at c equaling the limit value. Five examples are given and categorized as continuous, discontinuous with removable discontinuities, or discontinuous with nonremovable discontinuities. Removable discontinuities can be resolved by redefining the function value at c, while nonremovable discontinuities cannot be resolved.
Chapter 5 Definite Integral Topics Review
This document discusses several topics in calculus including RAM theory for approximating functions, computing definite integrals using formulas and a TI-89 calculator, properties of definite integrals like additivity and the mean value theorem, the fundamental theorem of calculus in two parts, and the trapezoidal rule for approximating definite integrals which overestimates for increasing functions and underestimates for decreasing functions.
Lesson 4.3 First and Second Derivative Theory
1) To determine if a function is increasing or decreasing, examine the sign of the first derivative f'(x) at points around any critical points. A change from positive to negative indicates a relative maximum, and a change from negative to positive indicates a relative minimum.
2) A curve is concave up if the second derivative f''(x) is positive, and concave down if f''(x) is negative. To determine concavity, examine the sign of f''(x) around any points where it is zero or undefined. A change in sign indicates an inflection point where the concavity changes.
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2.6 Related Rates
- 2. Related Rates Def: Related Rates are the rates of change of two or more related variables that are changing with respect to time. The basic idea behind related rates is that each of the variables is now a function of time, or t. When we differentiate, we will do so with respect to t. Some of the variables will not be changing with respect to t. These variables are like constants, and their derivatives with respect to t is 0.
- 3. Solving Related Rates Step 1: Identify the changing quantities, possibly with the aid of a sketch. Step 2: Write down an equation that relates the changing quantities. Step 3: Differentiate both sides of the equation with respect to t. Step 4: Solve for the required quantity, and substitute the given values in the derived equation.
- 4. Example 1: One day Ms. Henry arrived home late from work because she was tutoring calculus students. Her geriatric dog, Kelty, managed to hold herself for 13 hours, but when Ms. Henry let her out, and she noticed the area of Kelty’s circular doggie puddle was growing at a rate of 12 square inches per second. a) How fast was the radius growing at the instant when it equaled 20 inches? b) How fast was the circumference changing at the same time?
- 5. Example 2: Ms. Henry accidentally hit a calculus student in the head with a snowball. In retaliation, the student formed a snowball to return the fire, and as it was being formed, it was melting. If the snowball is melting at the rate of 10 cubic inches per minute, at what rate is the radius changing when the snowball is 2 inches in radius? http://www.mathopenref.com/calcsnowballproblem.html
- 6. Example 3: A 25 foot ladder is leaned against the wall of a building. Because it is not placed on stable ground, the base of the ladder begins slipping away from the wall at a rate of 2 feet per second. a) How fast is the top moving down the wall when the base of the ladder is 7 feet from the wall? b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. http://astro.ocis.temple.edu/~dhill001/relatedrates/ladderdemo.gif
- 7. Example 4: Dry sand is loaded onto a conveyor belt. The conveyor belt moves sand up at a fixed rate, and it spills off to form a conical pile, with the height equal to the radius. How fast is the height increasing when the height is 3 feet and sand is being added at a rate of 2 cubic feet per minute? http://astro.ocis.temple.edu/~dhill001/relatedrates/sandpiledemo.mov