MATH 220 Final: MATH 220 Amherst S18Final Exam Final
15 Feb 2019
School
Department
Course
Professor
Mathematics 106 FINAL EXAMINATION May 7, 2012
200 Points 3 Hours SHOW ALL WORK
1. [35 Points] Differentiate the following functions of x:
a. ln(x) sin(x)
b. x
cos3(x)
c. Zx
0
1
t+etdt
d. ln(√ex) + ln 2 (Your answer should be as simple as possible.)
e. (sec x)x
2. [35 Points] Evaluate the following integrals:
a. Ztan2(x) sec2(x)dx
b. Zln 4
ln 2
(ex)2dx
c. Zex
2 + exdx
d. Z√π/3
0
3xcos(x2)dx
e. Z(x2+ 1)2
x3dx
3. [10 Points] Find the equation of the line tangent to the curve y= sin(ex) at the point
where the x-coordinate is ln( π
4).
4. [10 Points] Assume that the amount of sand in a pile increases at a rate of 90t/(9+t2)2
cubic feet per minute. How much sand was added during the time period 0 ≤t≤4
minutes?
5. [25 Points] Consider the function f(t) = 3 sin(π
2t).
a. Determine the amplitude, period and frequency of this function.
b. Draw the graph of this function for one complete period.
c. Using only the graph of part b, explain whether the integral R4
13 sin(π
2t)dt is
positive, negative, or zero.
d. Approximate R4
13 sin(π
2t)dt using a Riemann sum with n= 3 and the left end-
point of each interval. (It is a bad approximation.)
e. Compute the integral R4
13 sin(π
2t)dt exactly.
1
Document Summary
Show all work: [35 points] di erentiate the following functions of x, ln(x) sin(x) b. x cos3(x) 0: [10 points] find the equation of the line tangent to the curve y = sin(ex) at the point where the x-coordinate is ln( . 4 ): [10 points] assume that the amount of sand in a pile increases at a rate of 90t/(9+t2)2 cubic feet per minute. How much sand was added during the time period 0 t 4 minutes: [25 points] consider the function f (t) = 3 sin( . 2 t) dt using a riemann sum with n = 3 and the left end- Do not evaluate the integral: express the area using an integral with y as the independent variable. Do not evaluate the integral: evaluate one of the integrals from parts a and b, [15 points] simplify as much as possible.
