Math 191   Calculus 1  aacchead.jpg (10305 bytes) 
Brief Study Guide for Chapter 7

Preview of Chapter 7
In this chapter we consider the topic of inverse functions, introduce 22 new functions and use l'Hopital's rule.  The 22 new functions consist of
  2 logarithmic functions,
  2 exponential functions,
  6 inverse trigonometric functions,
  6 hyperbolic functions and
  6 inverse hyperbolic functions. (We actually don't cover these last 6 functions.)
For each one of these functions, mathematicians want to know about or how to do roughly a dozen things, referred herein as the The "Dozen" Aspects of New Functions
As you are introduced to each of the 16 functions that we consider, analyze it against the list above.

Calculus 1        Lesson 7.1       Inverse Functions
The following topics are the most important. Typical exercises from Calculus, 4th edition, by James Stewart, are assigned at the end of each objective.
1. Do the following symbolically, numerically and graphically where applicable: Given a function f over a domain,
a. determine whether it is one-to-one; P391#3,5,7,9,11,15
b. find its inverse, g (or f -1), if it exists; P391#25,27,29,31,33
c. give the domain and range of f and g; P392#35,37
d. find the derivative of f and g and the connection between them; P392#39,41
e. give the equation of the line tangent to the functions f and g at corresponding places; 
f. graph the function f and its inverse g on the same set of axes. P392#35,37

Calculus 1        Lesson 7.2*      Natural Logarithms

Note: Sections 7.2*, 7.3* and 7.4* are printed on the gray pages of Stewart following sections 7.2, 7.3 and 7.4.
1. Define the natural logarithm function. Text
2. Analyze the graph of f(x) = ln x using f, f ', and f ''. Text
3. Use algebraic properties of ln x. P428#1,3,5,7
4. Graph expressions involving ln x without a calculator. P428#9,11
5. Differentiate functions involving ln x including implicit differentiation. P428#13,15,21,25,27
6. Give the equation of the tangent line of a function involving ln x. P428#41
7. Evaluate integrals involving the natural logarithm. P429#59,61,65
8. Use logarithmic differentiation. P429#55,57
Maple Assignment: none

Calculus 1        Lesson 7.3*      Natural Exponentials

Note: Sections 7.2*, 7.3* and 7.4* are printed on the gray pages of Stewart following sections 7.2, 7.3 and 7.4.
1. Give the definition of natural exponentiation. Text
2. Solve equations involving natural exponential and logarithmic expressions. P435#3,5,7,9
3. Graph expressions involving ex without a calculator. P435#17,21
4. Differentiate functions involving exponential and logarithmic functions including the chain rule and implicit differentiation. P435#29-41 odd
5. Give the equation of the tangent line of a function involving ex. P435#43
6. Solve word problems involving logarithms and exponentials. P436#53,55
7. Analyze the bell shaped curve from statistics. P436#66
8. Integrate expressions involving the natural exponential function. P436#67,69,71,73
Maple Assignment: See Maple Assignment

Calculus 1    Lesson 7.4*     General Exponentials and Logs

Note: Sections 7.2*, 7.3* and 7.4* are printed on the gray pages of Stewart following sections 7.2, 7.3 and 7.4.
1. For the general exponential and logarithm functions, do all of the following:
1.a define them; Text
1.b. evaluate; P445#7,13
1.c graph; P445#11,12,15
1.d finds limits; P445#21,22
1.e differentiate using chain rule; P445#23-35 odd
1.f integrate; P446#41,42,43,44
2. Compare the growth of the power function versus the exponential function. P445#19
3. Find the equation of the tangent line of a curve involving logarithms and exponentials. P445#39
4. Find the area under the curve involving logarithms and exponentials. P446#43
5. Give an overview of the development of logarithm and exponential functions. Text: sections 7.2*-7.4*
Maple Assignment: none

Calculus 1     Lesson 7.6     Inverse Trigonometric Functions
1. For the inverse trig functions, do all of the following: 
1.a define; Text
1.b. evaluate; P461#1,5,11
1.c graph; P461#15
1.d finds limits; P461#45
1.e differentiate using chain rule; P461#23-33 odd
1.f integrate; P462#59-67
2. Derive the formula for the derivative of an inverse trig function.
(sin-1, cos-1 or tan-1) . P461#19
3. Find the equation of the tangent line of a curve involving inverse trig functions. P461#40
4. Find the area under the curve involving inverse trig functions. P462#73
5. Solve a related rates word problem involving inverse trig functions. P462#49
Maple Assignment:  Combined with section 7.6 maple worksheet.

Calculus 1     Lesson 7.7     Hyperbolic Functions
1. For the hyperbolic functions, do all of the following: 
1.a define; Text
1.b. evaluate; P468#1,3
1.c graph; P469#22
1.d finds limits; P469#23a,e
1.e differentiate using chain rule; P469#31,37,43,47
1.f integrate; P470#55,56,59,61
2. Derive the formula for the derivative of a hyperbolic function.  P469#24a,b
3. Find the equation of the tangent line of a curve involving hyperbolic functions. (no specific examples in text)P470#53
4. Find the area under the curve involving hyperbolic functions. (no specific examples in text)
5. Solve a catenary word problem P469#49
6. Explain why trig functions are often called circular functions and hyperbolic functions are so named.
Maple Assignment: See Maple Assignment

Calculus 1     Lesson 7.8     l'Hôpital's Rule
1. Name the seven indeterminate forms. Text
  P470-478 wpdoc0{image0}.gif (1365 bytes) These indeterminate forms are discussed.
2. Determine whether l'Hôpital's rule applies and, if so, apply it to find limits of the seven indeterminate forms. P478#5-61 every other odd.