MATH 4B – STUDY GUIDE FOR CHAPTER 7
1. Find the derivatives of log or exponential functions; the problem(s) will require the chain rule
and possibly the product or quotient rule.
The bases may be e or some other value.
See the examples in #8 below
2. Differentiate the natural log
of an expression involving products, quotients and powers.
To receive credit, you must use properties of logs to expand the
expression completely before differentiating
Ex: Find 
It is not necessary to add fractions to finish.
3. Perform logarithmic
differentiation.
Ex: Find the
derivative of xsin(x) by logarithmic
differentiation
4. Calculate an integral. (Integrand involving trig functions (sin, cos, tan or cot), exponential or log functions with base e
or some other base; careful u- substitution
will be needed).
Ex: 
5. Find the volume of the solid of
revolution.
Ex: Solid formed
when the region bounded by y=e−2x, y=0, x=0, x=ln(3) is
revolved about the x-axis
6. (a) Find the derivative of y= [inverse trig or hyperbolic function] using the inverse function procedure.
(b) Write the
derivative as a purely algebraic function of x.
(c) Write the corresponding integration formula.
Ex: a. Compute the derivative of y=cosh-1x using the inverse function theorem.
b. Write the derivative as a purely algebraic function
of x. (you will
have to use the identity that relates cosh2x and sinh2x)
c. Use the result of Part b to integrate 
Hint: Start with a
u-substitution.
7. Calculate an integral [Function requiring long division, splitting
a fraction into a sum of two fractions, and/or u-substitution; resulting integral may require an inverse
sine, inverse tangent, natural log, or power rule]
Ex: 6. Integrate 
Hint: start by dividing.
8. If f(x) = _______________
[function involving an exponential
or log function]
Ex: 
or f(x) = x2ln(x)
·
Calculate the first and second derivatives. (Expect to use the product rule or quotient
rule).
·
Identify all intervals where f
is increasing and decreasing.
·
Locate and label all extrema
·
Locate and label all inflection points.
·
Identify the interval(s) where the function is (a)
increasing, (b) decreasing, (c) concave up,
(d) concave down
9. Solve a differential equation (initial
value problem) with a given initial
condition, or determine whether a given expression is a solution to a
differential equation.
Ex: Solve the
initial value problem 
if y(0)=2 and if (3+5y) is positive.
10. Evaluate the limit using L’Hôpital’s rule, if applicable: 
.
The form may be 0/0, ∞/∞, 0 times ∞, 1∞,
00, or ∞0 (Remember, the last 3 require a logarithm
technique)
Be able to recognize whether L'Hopital's rule
applies: is the form in the problem
"indeterminate" or not?
11. Modeling problem. Here is an example: Suppose an observer is watching an aircraft
as it approaches. The aircraft is at an
altitude of 6 miles and is flying directly toward the observer. As the aircraft approaches, its angle of
elevation q (the angle between the ground and the
line of sight to the aircraft) will change
·
Find an expression for the angle of elevation q in
terms of the horizontal distance x to the aircraft.
·
Find an expression for 
, the rate at which the angle of elevation is changing. Remember the chain rule.
·
If the angle of elevation is changing at a rate of 2
degrees per second at the moment when the angle of elevation is 45°,
what is the aircraft’s speed 
? (Convert degrees to
radians!)
Similar problems:
How fast is a balloon rising? How
fast is a rocket climbing?
12. Use the definition of
specific hyperbolic functions (example: sinh x = 
) to evaluate a more complicated hyperbolic expression (write
it in terms of exponentials, then simplify;
example: sinh(2
ln x)
13. Prove or identify an
identity involving hyperbolic functions
Ex: a. Write the definitions of sinh(x) and cosh(x)
b. Using the definitions, multiply and simplify the expression 2sinh(x)cosh(x).
c. What hyperbolic expression does the result represent?
14. Differentiate and/or
integrate an expression including hyperbolic functions.
Ex: Integrate 
Find dy/dx when y = cosh3(5x)
Return to: Merced College; Don Power
Updated 09/14/07 by Don Power