APPM 1350 Midterm 3 Fall 2011
On the front of your bluebook, please write: a grading key, your name, student ID, section, and instructor’s
name (Chang, Curry, Dougherty, Guinn, Nelson). This exam is worth 100 points and has 7 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. (18 points) Find dy/dx for each of the following functions.
(a) y=Zx3
x
t3
t2+ 1 dt (b) y=xln(x2+ 10) (c) y=√9x−1
(2x3+ 1)2(x−2)3(Use logarithmic
differentiation.)
2. (30 points) Evaluate the following integrals. Be sure to show all work.
(a) Z5x2
√x3−2dx (c) Zx√3x−1dx (e) Zsec22θtan 2θ dθ
(b) Z3
−3
x6sin x
x2+ 4 dx (d) Z0
−1
3dx
3x−2
3. (12 points) For each of the following, write TRUE or FALSE (not just T or F).
No justification is necessary.
(a) Zxcos x dx =xsin x+ cos x+C.
(b) If fis continuous on [−1,1] then there is a
cin [−1,1] such that f(c) = 1
2Z1
−1
f(t)dt.
(c) Zb
a
f(−x)dx =Z−b
−a
f(x)dx
(d) lim
n→∞
n
X
i=1 r2i
n2
n= 2 Z1
0
√x dx
4. (12 points)
(a) Evaluate the Riemann sum Rn=
n
X
i=1 i
n21
n. Express your answer in terms of n.
(b) Use the answer for part (a) to evaluate lim
n→∞
Rn.
(c) Express lim
n→∞
Rnas a definite integral.
5. (12 points) Let g(x) = Zx
2
f(t)dt, 2≤x≤10,
where fis the function defined on [2,10] whose graph is shown at
right. No justification is necessary for the following questions.
(a) On what intervals is gdecreasing?
(b) At what values of xdoes ghave local minimum values?
(c) Where does gattain its absolute minimum value?
(d) On what intervals is gconcave down?
1
3
5
7
9
11
t
-8
-4
4
y
yáfHtL
6. (8 points) Use Newton’s Method to estimate the roots of x3−x2−x+ 1 = 0. Use an initial approximation
of x1= 0 and find x2and x3. What can you conclude from your answers?
7. (8 points) Consider the function g(x) = 1
x2+ 2, x > 0.
(a) Show that gis one-to-one and thus invertible.
(b) Find the inverse g−1(x).
(c) State the domain and range of g−1(x).