Math 1B Old Chapter 5 Test Problems
1. Simplify each expression (hint: none of these is difficult):
a.
b.
c.
2.
Evaluate the definite integral . hint:
.
3.
Find an elementary antiderivative for .
4.
Evaluate .
Hint: Use integration by parts to
establish a recursion formula. Keep in
mind that this is a definite integral.
5. Show
that the integral is convergent and its value is between 0.5
and 1.
6. Determine
whether the integral is convergent or divergent.
7. Let
,
where f is the function whose graph is shown below:
a.
Make a table of values showing the value of at x = 0, 2, 3 and 4
b.
Sketch a graph for on
.
c.
What are the global maximum and the global minimum for ?
8. Write
a definite integral which is equivalent to this limit of a Riemann sum:
9. Simplify each expression (hint: none of these is very difficult):
a.
b.
c.
10. Evaluate
the definite integral .
11. Find an
elementary antiderivative for .
12. Evaluate .
Hint: Use integration by parts to
establish a recursion formula. Keep in
mind that this is an improper integral.
13. Use
comparison to show that converges to a value between 1/4 and 1/2.
14. Determine
whether the integral is convergent or divergent.
Rewrite the integral using the substitution
. What is du? What are the new bounds of integration? Evaluate the integral.
15. Use substitution to simplify the following definite integrals. For each, explicitly state what the components of your substitution are (on separate paper):
a. u =
du = ____________?
b.,
where
u = _____ ? du = ___________?
c. (hint:
) u
= _____ ? du = ___________ ?
16. Use integration by parts to compute the following definite integrals. For each, explicitly state what the components of your substitutions are (on separate paper):
d.
e.
17. Use a
trigonometric substitution to evaluate .
18. Let . Do integration by parts twice so that I recurs on the right side of your
equation. Solve the equation for I.
19. Let . Prove that
.
20. Use the
Fundamental Theorem of calculus to find if
21. Show how to
use the methods of partial fractions and completing the square to compute . Hint: the antiderivative may involve two
logarithms and an arctan function.
22. A
nonnegative function f is called a probability density function if .
a. Find
a value of A so that is a probability density function.
b. Is
there a value of A so that is a probability density function? Explain.
23. The Gamma
Function is defined by
c. Find
d. Find
24. Consider
the integral for the arclength of a quarter of the unit circle: .
e. Find
the exact value for this integral using the fundamental theorem.
f.
Approximate this arclength using a trapezoidal sum with
n = 4 equal subintervals.
g. Approximate
this integral using a midpoint sum with n = 4 equal subintervals.
h. Approximate
this integral using a Simpson sume with n = 8 equal subintervals.
i. Why is the estimated upper bound on the error not particularly useful for these approximations?