APPM 1350 Midterm 3 Spring 2012
On the front of your bluebook, please write: a grading key, your name, student ID, section, and
instructor’s name (Chang or Guinn). This exam is worth 100 points and has 9 questions. Show all work!
Answers with no justification will receive no points. Please begin each problem on a new page. No notes,
calculators, or electronic devices are permitted.
1. (15 points) Differentiate the following functions. Leave your answers unsimplified.
(a) y= ln(sin(ln x)) (b) y=Z2/x
2x
1−t
1 + t2dt
(c) y=4√x3−3
x3
√x+ 7 (use logarithmic differentiation)
2. (18 points) Evaluate the following integrals.
(a) Z(1 −tan 3t) sec23t dt (b) Zcot θ
4dθ
(c) Z1
0
10√x
(1 + x3/2)2dx
3. (8 points) The function f(t) = cos t+t−t2has only one maximum value. Estimate the value of t
where the maximum value of f(t)occurs using one iteration of Newton’s Method. Use t1= 0 as an
initial approximation.
4. (15 points) Consider the definite integral Z2
0x2−2xdx.
(a) Estimate the integral by dividing the interval [0,2] into 4equal subintervals and using right
endpoints for evaluation (find R4).
(b) Estimate the integral by dividing the interval [0,2] into nequal subintervals and using right
endpoints for evaluation (find Rn). Your answer should be in sigma notation.
(c) Simply the expression for Rnthat you found in part (b). Your answer should no longer be in
sigma notation and should be in terms of only n. Leave your answer unsimplified.
(d) Find the exact value of the integral by evaluating lim
n→∞ Rnusing your answer from part (c).
(e) Check your work in part (d) by evaluating the definite integral directly using the Fundamental
Theorem of Calculus (Evaluation Theorem).
