IB Maths.Second derivative. Gradient as rate of change
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This document discusses the concept of derivatives and rates of change. It provides examples of calculating first and second derivatives of functions, using derivatives to find velocity and acceleration from displacement-time graphs, and using derivatives to determine rates of change in various contexts like population growth. The key ideas are that the derivative represents the rate of change of one variable with respect to another, and it can be used to calculate velocities, accelerations, and other instantaneous rates of change from functional relationships.
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InstructorCourseBook Bittinger Calculus and ItsApp.docx
Instructor:
Course'
Book: Bittinger: Calculus and Its
Applications, 10e
Compute the difference quotient
answer as much as possible.
I
for the function f (*) : ----=' Simplifu yourx*9
f(x+h)-f(x)
h
f (x+h)-f (x) _
h (x+9)(x+9+h)
-16x*3-
-1
2.
Compure the difference quotient
"IJ+-!t
for the function f(*) : - 4x2 - 3x_ 2'
Simphfy your answer as much as possible'
f (x+h)-f (x) _
Use the graph to estimate the average rate of
change of the percentage of new employees
from 1994 to 2000, from 2000 to 2006, and from
1994 to 2006.
What is the average rate of change of the
percentage of new employees from 1994 to
2000?
Yo Per Year
('l--vpe an int"egerr or a decimal rounded to trvo
decimal places as necded')
What is the average rate of change of the
percentage of new employees from 2000 to
7006?
o/o
Pet Yeat
('Iype an integer or a dccimai rounded Io t\^'o
decirnal piaces as neeiled")
What is the average rate of change of the
percentage of new employees from [994 to
2006?
Yo Per Year
(T-vpe an itrteger or a decimal roundeii to two
clecimal piaccs as neecled )
aJ.
0
o
Ernployment
Page 1
Student:
Date: j
Time:
Assignment: HW 1.3
Instructor:
Course:
Book: Bittinger: Caleulus and Its
Utitty is a type of function that occurs in
econ;mics. When a con$rmer feceives x units of
a certain produc! a certain amourt of pleasure,
or utility-U, is derived. The follouring is a graph
of a utiiity firnction- Find the average rate of
change ofU as x changes &om 0 to 1; &om I to
2; from 2to3; from 3 to4-
Find the average rate ofchange ofU as x
changes from 0 to 1'
tr (simptiff your answer.)
Find the average rate of change of U as x
changes from I to 2-
I (si*priff Your answer.)
Find the average rate ofchange ofU as x
changes from 2 to 3.
I (si-priff Your answer.)
Find the average rate of change of U as x
changes from 3 to 4.
[ 1si-prify Your answer-)
Page2
Student:
Date:
Time:
Assignment: }{W 1.3
Applications, 10e
ia-
Instructor: {
Course:
Book: Bittinger: Clalculus and Its
Applications, 10e
Suppose that a fashion company determines that the cost,
in dollars, of producing x
cellphone-sunglasses is given by C(x): - 0'05x2 + 55x'
c(301) - c(300)
r;,ayfrffi.andinterprerthesignificanceotthisresulttothecompany.
c(30l) - c(3oo)
rind -- ,['1 _ 369 -.
C(301) - C(300) - {Simplify your ansr,ver.)
301 - 300
Interpret the result'
It represents the average cost to produce one item'
It represents the average cost per item to produce 300 items"
It tells us the average cost to produce 301 items'
lt represents the average cost to produce one more item after making 300
items'
Page 3
Student:
Date:
Time:
Assignment: HW 1.3
5.
6. The home rumge, in hectares,
of a carnivorous mammal weighing w grams can be approximated
by
H(w):0.1lwr'36'
a) Find the average rate at which a carnivorous mammal's
horne range increases as the animal's
weight grows from 350 g to 500 g'
H(3oo) - H(2oo)
b) Find -ff;ri;-. ar.
3.1 limit definition of the derivative
This document is a lesson plan on defining the derivative using the limit definition. It includes objectives, an agenda, warm-up problems, notes on the limit definition of the derivative with examples, and guided practice problems. The lesson covers finding the slope of a curve at a point, writing equations of tangent lines, and evaluating limits used to define the derivative. Students are assigned homework problems involving finding derivatives and writing tangent line equations.
Application of Integration
This document discusses various applications of integration, including:
1) Velocity and distance problems involving motion along a line
2) Calculating the area of different regions bounded by curves
3) Finding volumes of solids obtained by rotating areas about axes
4) Determining arc lengths and lengths of curves
5) Calculating velocity and acceleration vectors of objects moving along planar curves
6) Applying integrals to problems involving work, average values, forces, and other rates of change.
Application of Integration
This document discusses various applications of integration, including:
1) Velocity and distance problems involving motion along a line
2) Calculating the area of different regions bounded by curves
3) Finding volumes of solids obtained by rotating areas about axes
4) Determining arc lengths and lengths of curves
5) Calculating velocity and acceleration vectors of objects moving along planar curves
6) Applying integrals to problems involving work, average values, forces, and other rates of change.
Application of Integration
This document discusses various applications of integration, including:
1) Velocity and distance problems involving motion along a line
2) Calculating the area of different regions bounded by curves
3) Finding volumes of solids obtained by rotating areas about axes
4) Determining arc lengths and lengths of curves
5) Calculating velocity and acceleration vectors of objects moving along planar curves
6) Applying integrals to problems involving work, average values, forces, and other rates of change.
Application of Integration
This document discusses various applications of integration, including:
1) Finding distances, velocities, and areas involving motion along a line
2) Calculating areas of regions bounded by curves
3) Computing volumes of solids obtained by rotating regions about axes
4) Determining arc lengths, velocities, and accelerations for planar curves
It provides examples and problems for each application, as well as frequently asked questions.
Application of Integration
This document discusses various applications of integration, including:
1) Velocity and distance problems involving motion along a line
2) Calculating the area of different regions bounded by curves
3) Finding volumes of solids obtained by rotating areas about axes
4) Determining arc lengths and lengths of curves
5) Calculating velocity and acceleration vectors of objects moving along planar curves
6) Applying integrals to problems involving work, average values, forces, and other rates of change.
Application+of+Integration
This document discusses various applications of integration, including:
1) Velocity and distance problems involving motion along a line
2) Calculating the area of different regions bounded by curves
3) Finding volumes of solids obtained by rotating areas about axes
4) Determining arc lengths and lengths of curves
5) Calculating velocity and acceleration vectors of objects moving along planar curves
6) Applying integrals to problems involving work, average values, forces, and other rates of change.
Application of Integration
This document discusses various applications of integration, including:
1) Finding distances, velocities, and areas involving motion along a line
2) Calculating areas of regions bounded by curves
3) Computing volumes of solids obtained by rotating regions about axes
4) Determining arc lengths, velocities, and accelerations for planar curves
It provides examples and problems for each application, as well as frequently asked questions.
Application of Integration
This document discusses various applications of integration, including:
1) Velocity and distance problems involving motion along a line
2) Calculating the area of different regions bounded by curves
3) Finding volumes of solids obtained by rotating areas about axes
4) Determining arc lengths and lengths of curves
5) Calculating velocity and acceleration vectors of objects moving along planar curves
6) Applying integrals to problems involving work, average values, forces, and other rates of change.
Application of Integration
This document discusses various applications of integration, including:
1) Velocity and distance problems involving motion along a line
2) Calculating the area of different regions bounded by curves
3) Finding volumes of solids obtained by rotating areas about axes
4) Determining arc lengths and lengths of curves
5) Calculating velocity and acceleration vectors of objects moving along planar curves
6) Applying integrals to problems involving work, average values, forces, and other rates of change.
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6) equations with exponentials & logarithms
This document provides worked examples for solving various logarithmic and exponential equations. It begins with an introduction stating the learner will be able to solve exponential equations of different bases, logarithmic equations, and exponential equations of the same base. It then shows 7 worked examples of solving equations involving logarithms and exponents, with the conclusion being about expressing exponential powers in the same base when possible.
7) change of base
This document discusses logarithmic equations and calculations. It provides instructions on how to rewrite logarithmic equations without using logarithms, solve simultaneous logarithmic equations, calculate logarithms to specific bases and numbers of significant figures, change logarithmic bases, and solve logarithmic equations. The document aims to teach students how to perform various operations with logarithms including rewriting, solving, calculating, and changing bases.
5) logarithms graphs
This document provides an overview of exponential and logarithmic functions:
1) It explores properties of exponential functions with the forms y = ax, where a > 1 represents exponential growth and 0 < a < 1 represents exponential decay. Key aspects like asymptotes and y-intercepts of these graphs are discussed.
2) Tables are completed and graphs are drawn to represent exponential growth and decay functions.
3) The natural logarithm function ln(x) is introduced and its graph is sketched using a calculator. Common features that logarithmic graphs share with exponential graphs are described.
4) Examples of simplifying logarithmic expressions and solving logarithmic equations are presented.
4) logarithms laws
This document discusses logarithmic laws and their proofs. It contains:
- Exercises to verify logarithmic laws like loga(xy) = loga(x) + loga(y) and nloga(x) = loga(x)n
- Proofs of these laws using the definition of logarithms and index form
- More exercises for students to solve involving simplifying logarithmic expressions using these laws
The goal is for students to understand and be able to apply the key logarithmic laws through worked examples and practice questions.
3) logarithm definition
This document introduces logarithms and how to use them to solve exponential equations. It defines logarithms as the power to which a base number must be raised to equal the value being logged. Examples are provided of writing numbers and powers as logarithms in different bases. The basics that the logarithm of a number to itself is 1 and the logarithm of 1 or 0 is 0 or undefined are covered. Students are directed to an online worksheet and book exercises to practice evaluating and using logarithms.
2) exponential growth and decay
This document discusses exponential functions and their properties. It explores exponential growth and decay through graphs of functions like y=2^x and y=0.5^x. It shows that as x increases, exponential growth functions approach infinity, while decay functions approach zero. The document also introduces the irrational number e as the most important base for modeling continuous growth and decay. It shows how the function A=Ce^rt models continuous compound interest as the compounding period approaches infinity.
1) exponential equations
This document discusses exponential functions and equations. It explores the properties of exponential functions by examining their graphs and understanding exponential growth and decay. It also covers how to solve problems leading to exponential equations by learning about bases, exponents, powers, and index rules. The document teaches how to evaluate, write in different bases, identify exponential equations, and solve exponential equations. It includes exercises to solve from the attached document on exponential equations and functions.
6) simultaneous and problems
This document discusses finding the intersection points between a line and parabola by solving simultaneous equations, finding the value of k that makes a line tangent to a parabola, and finding the maximum area of a rectangle with a perimeter of 28 cm.
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This document discusses solving quadratic equations by different methods such as factorisation, using the quadratic formula, and completing the square. It explains how to determine the number of solutions a quadratic equation has based on the sign of its discriminant, namely:
- If the discriminant is positive, there are two real roots
- If the discriminant is zero, there is one repeated root
- If the discriminant is negative, there are no real roots.
Examples are provided to illustrate calculating the discriminant and determining the number and type of roots.
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This document discusses quadratic functions and their properties. It explains how to express quadratics in different forms like factored, general, and vertex form. It provides examples of finding the zeros, line of symmetry, and vertex of various quadratic functions. Students are asked to work through examples of sketching graphs, factorizing expressions, solving equations, and rewriting functions in different forms to investigate the properties of quadratic functions and their relationships between algebraic and graphical representations.
3) quadratic vertex form
The document discusses quadratic functions and their graphs. It aims to explore the properties of quadratic functions and express them in different forms, including vertex form. It provides examples of writing quadratic functions in vertex form given the vertex, as well as writing the function in general form given its vertex. The key points are that the vertex form of a quadratic function y = a(x - h)2 + k shows the vertex is (h, k) and the line of symmetry is x = h. Converting between the vertex and general forms allows finding the maximum point of a quadratic function.
2) quadratics gral form
This document discusses properties of quadratic functions and their graphs. It explores expressing quadratic functions in different forms, sketching graphs of basic quadratic functions, and investigating the effects of changing coefficients on the graph. The key aspects covered are the relationship between the coefficients and the graph properties like the vertex, line of symmetry, and y-intercept. Examples of matching functions to graphs and writing functions in standard form are provided.
1) quadratic functions
The document discusses quadratic functions and their graphs. It explores expressing quadratic functions in different forms, and transforming quadratic functions and their graphs through translations, stretches, and other changes. Key aspects covered include the vertex and line of symmetry of parabolas, and how changing the coefficients affects the graph shape and position. Examples of specific quadratic functions are analyzed and graphed.
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This document discusses two methods for solving simultaneous equations: the elimination method and the substitution method. It provides examples of using the elimination method to solve several systems of simultaneous equations, showing the steps to eliminate one variable and solve for the other variables. The document concludes by providing two word problems as examples and assigning practice problems from the book.
Y9 algebra 1 simultaneous equations by substitution method
The document provides examples of solving simultaneous equations by substitution. It includes examples of setting up two equations representing relationships between variables, substituting one equation into the other to isolate a single variable, and solving for the value of that variable. It then uses the solved variable to find the value of the other variable.
Y9 algebra 1 Change of Subject
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Y9 algebra 1 Inequalities
This document provides an overview of inequalities and how to solve them algebraically. It discusses representing inequalities on number lines, and the rules for adding, subtracting, multiplying, and dividing terms when solving inequalities, such as changing the sign of the inequality when multiplying or dividing by a negative number. Examples are provided of solving inequalities algebraically and representing the solutions on number lines.
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- Expanding brackets
- Multiplying algebraic expressions
It then provides examples of simplifying algebraic expressions, expanding brackets, finding perimeters and areas of shapes, evaluating expressions, and converting between temperatures in Fahrenheit and Celsius.
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This document provides an overview of solving linear equations, including equations involving brackets, fractions, and changing the subject of a formula. It presents several example equations and step-by-step solutions showing how to isolate the variable. Students are reminded to check their answers by substituting them back into the original equation. The document also introduces solving equations with fractions using the lowest common denominator approach.
Ib maths sl chain rule
The document discusses applying the chain rule to find derivatives of composite functions. It provides examples of identifying composite functions and using the chain rule formula to take derivatives. It lists practice problems for students to find derivatives of various composite functions using the chain rule, including functions with exponents like (x3 + 1)8, as well as trigonometric functions like y = sin(4x).
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