Math 1B Chapter 5 Test (part 1)
Fall ’06
Name________________________
Directions: Show all work for credit. Write responses on separate paper. Do not use a calculator.
1.
Find the approximating sum for with n
= 2
a. Assuming subintervals of equal length, find the value of Δt and ti.
b. Using right endpoints as sample points in a Riemann sum with subintervals of equal length.
c. Using left endpoints as sample points in a Riemann sum with subintervals of equal length.
d. Using midpoints as sample points in a Riemann sum with subintervals of equal length.
e. Using a trapezoidal sum.
f. Using the Simpson sum.
2.
Use the definition of the definite integral to evaluate
as a limit of Riemann sums. Do not use the evaluation theorem. Recall that
,
and
3. Evaluate the integral. Clearly indicate any substitution, or integration by parts.
a.
b.
c.
d.
4.
Consider the function whose graph is shown:
a. Where does g(x) have a maximum? A minimum?
b. Where does g have an inflection point?
c.
Sketch an approximation for the graph of on the interval
5.
Evaluate . Use integration by parts twice and show your
method in detail.
Math 1B Chapter 5 Test (part 2) Name_______________________________
Directions: Show all work for credit. You may need to use a calculator for some problems, but for those problems where a calculator is not really needed, show the method without resorting to a calculator.
6.
Consider approximating sums for with n
= 20 subintervals of equal length.
a.
Write expressions for Δt, ti., ti-1 and
b.
Approximate to 6 significant digits the midpoint sum M20 and its error EM.
c.
Approximate to 6 significant digits the trapezoidal
sum, T20, and its error ET.
d.
Approximate to 6 significant digits the Simpson sum, S40, and its error ES.
7.
Consider the integral
a.
Find the partial fraction expansion of the integrand.
b.
Simplify the antiderivative.
8.
Use the substitution to evaluate
.
9.
Use comparison to show that the integral is convergent.
10. Determine
how large a has got to be so that .
Math 1B Chapter 5 Test Solutions
Fall ’06
1.
Find the approximating sum for with n
= 2
a.
Assuming subintervals of equal length, find the value
of Δt and ti.
and
.
b.
Use right endpoints as sample points in a Riemann sum
with subintervals of equal length.
c.
Use left endpoints as sample points in a Riemann sum
with subintervals of equal length.
d.
Use midpoints as sample points in a Riemann sum with
subintervals of equal length.
e.
Using a trapezoidal sum.
f.
Using the Simpson sum.
2.
Use the definition to evaluate as a limit of Riemann sums.
3. Evaluate the integral. Clearly indicate any substitution, or integration by parts.
a.
b.
c.
d.
= using
4.
Consider the function
whose graph was shown (the dotted curve in the graph below):
a.
has a maximum at about (1, 1.2) and a minimum at about (3, -1.2).
b. g has an inflection point at (2,0). It changes from concave down to concave up.
c.
Here’s a sketch for the graph of g on the interval ,
note the smoothing effect of integration:
5.
I = .
6.
Consider approximating sums for with n
= 20 subintervals of equal length.
a.
,
,
and
.
b.
The midpoint sum M20
can be set up like so.
As the screenshot below shows, the TI-200 calculates this to be 85.36, while
the true value is 85 + 1/3, meaning the error is EM
=
c.
Approximate to 6 significant digits the trapezoidal
sum, T20, and its error ET.
On the Voyage 200, I verified this with the average of left and right sums, so
Whence the error is ET
d.
Approximate to 6 significant digits the Simpson sum, S40, and its error ES.
The easiest way to compute the Simpson sum at this point is as the weighted
average This is an exact value for the integral, ES = 0.
7.
Consider the integral
a.
Find the partial fraction expansion of the integrand.
Equating coefficients quickly yields C
= 3, B = 1, A = 2, so
b.
Simplify the antiderivative.
- Use
the substitution to evaluate.
Now substitute
This is verified on the calculator:
- Use
comparison to show that the integral is convergent.
- Determine
how large a has got to be so
that .