Applications of the Derivative

Chapter Review

THINGS TO KNOW

4.1 Related Rates

Steps for solving a related rate problem (p. 256)

4.2 Maximum and Minimum Values; Critical Numbers

Definitions:

Absolute maximum; absolute minimum (p. 264)
Absolute maximum value; absolute minimum value (p. 264)
Local maximum; local minimum (p. 264)
Local maximum value; local minimum value (p. 264)
Critical number (p. 267)

Theorems:

Extreme Value Theorem If a function \(f\) is continuous on a closed interval \([a, b]\), then \(f\) has an absolute maximum and an absolute minimum on \([a, b]\). (p. 265)
If a function \(f\) has a local maximum or a local minimum at the number \(c\), then either \(f^\prime ( c) =0\) or \(f^\prime (c)\) does not exist. (p. 266)

Procedure:

Steps for finding the absolute extreme values of a function \(f\) that is continuous on a closed interval [a, b]. (p. 268)

339

4.3 The Mean Value Theorem

Rolle’s Theorem (p. 275)
Mean Value Theorem (p. 277)
Corollaries to the Mean Value Theorem

If \(f^\prime (x) =0\) for all numbers \(x\) in \(( a,b) \), then \(f\) is constant on \(( a,b) \). (p. 279)
If \(f^\prime (x) =g^\prime (x) \) for all numbers \(x\) in \(( a,b) \), then there is a number \(C\) for which \(f(x) =g(x) +C \hbox{ on } ( a,b). \) (p. 279)

Increasing/Decreasing Function Test (p. 279)

4.4 Local Extrema and Concavity

Definitions:

Concave up; concave down (p. 288)
Inflection point (p. 289)

Theorems:

First Derivative Test (p. 284)
Test for concavity (p. 288)
A condition for an inflection point (p. 290)
Second Derivative Test (p. 291)

Procedure:

Steps for finding the inflection points of a function (p. 290)

4.5 Indeterminate Forms and L'Hôpital's Rule

Definitions:

Indeterminate form at \(c\) of the type \(\dfrac{0}{0}\) or the type \(\dfrac{\infty }{\infty }\) (p. 299)
Indeterminate form at \(c\) of the type \(0\cdot \infty \) or \(\infty - \infty \) (p. 304)
Indeterminate form at \(c\) of the type \(1^{\infty }\), \(0^{0}\), or \(\infty ^{0}\) (p. 304)

Theorem:

L'Hôpital's Rule (p. 299)

Procedures:

Steps for finding a limit using L'Hôpital's Rule (p. 300)
Steps for finding \(\lim\limits_{x\rightarrow c}[f(x) ] ^{g(x) }\), where \([ f(x) ] ^{g(x) }\) is an indeterminate form at \(c\) of the type \( 1^{\infty }\), \(0^{0}\), or \(\infty ^{0}\) (p. 304)

4.6 Using Calculus to Graph Functions

Procedure:

Steps for graphing a function \(y=f(x).\) (p. 308)

4.7 Optimization

Procedure:

Steps for solving optimization problems (p. 319)

4.8 Antiderivatives; Differential Equations

Definitions:

Antiderivative (p. 329)
General solution of a differential equation (p. 331)
Particular solution of a differential equation (p. 332)
Boundary condition (p. 331)
Initial condition (p. 333)

Basic Antiderivatives See Table 6 (p. 330)

Antidifferentiation Properties:

Sum Rule: If functions \(F_{1}\) and \(F_{2}\) are antiderivatives of the functions \(f_{1}\) and \(f_{2}\), respectively, then \(F_{1}+F_{2}\) is an antiderivative of \(f_{1}+f_{2}\). (p. 330)
Constant Multiple Rule: If \(k\) is a real number and if \(F\) is an antiderivative of \(f\), then \(kF\) is an antiderivative of \(kf\). (p. 331)

Theorems:

If a function \(F\) is an antiderivative of a function \(f\) on an interval \(I\), then any other antiderivative of \(f\) has the form \(F(x) +C\), where \(C\) is an (arbitrary) constant. (p. 329)
Newton’s First Law of Motion (p. 336)
Newton’s Second Law of Motion (p. 336)

OBJECTIVES

Section	You should be able to...	Examples	Review Exercises
4.1	1 Solve related rate problems (p. 255)	1–5	1–3
4.2	1 Identify absolute maximum and minimum values and local extreme values of a function (p. 263)	1	4, 5
	2 Find critical numbers (p. 267)	2	6(a), 7
	3 Find absolute maximum and absolute minimum values (p. 268)	3–6	8, 9
4.3	1 Use Rolle’s Theorem (p. 275)	1	10
	2 Work with the Mean Value Theorem (p. 276)	2, 3	11, 12, 28
	3 Identify where a function is increasing and decreasing (p. 279)	4–6	23(a), 24(a)
4.4	1 Use the First Derivative Test to find local extrema (p. 284)	1, 2	6(b), 13(a)–15(a)
	2 Use the First Derivative Test with rectilinear motion (p. 286)	3	16
	3 Determine the concavity of a function (p. 287)	4, 5	23(b), 24(b)
	4 Find inflection points (p. 290)	6	23(c), 24(c)
	5 Use the Second Derivative Test to find local extrema (p. 291)	7–8	13(b)–15(b)
4.5	1 Identify indeterminate forms of the type \(\dfrac{0}{0}\) and \(\dfrac{\infty }{\infty }\) (p. 298)	1	41–44
	2 Use L'Hôpital's Rule to find a limit (p. 299)	2–6	45, 47, 49–52, 55
	3 Find the limit of an indeterminate form of the type \(0\cdot \infty,\) & \(\infty -\infty\), \(0^{0}\), \(1^{\infty }\), or \(\infty ^{0}\) (p. 303)	7–10	46, 48, 53, 54, 56
4.6	1 Graph a function using calculus (p. 308)	1–7	17–22, 25, 26, 27
4.7	1 Solve optimization problems (p. 318)	1–6	29, 30, 61–63
4.8	1 Find antiderivatives (p. 329)	1, 2	31–38
	2 Solve a differential equation (p. 331)	3, 4	57–60
	3 Solve applied problems modeled by differential equations (p. 333)	5–7	39, 40, 64