Math 242: Calculus 1
Exam 4
Name:
myWSU ID:
SCORE
/ 80
Read all of the following information before starting the exam:
• Show all work, clearly and in order, if you want to get full credit. I reserve the right
to take off points if I cannot see how you arrived at your answer (even if your final
answer is correct).
• Justify your answers whenever possible to ensure full credit. If you do use your calculator, explain all relevant mathematics.
• No notes or books are allowed.
• If you use more paper, please staple the pages to your test before you hand it in.
• Good luck!
1. (4 points) The graph of f is shown below. Shade the portion which corresponds to
Z 6
f (x) dx
2
(shade carefully; it’s not correct if the wrong region is shaded!)
7
6
5
4
3
2
1
−3
−2
−1
−1
1
2
3
4
5
6
7
8
−2
2. (6 points) Find the average value of the function f (x) = sec2 x on the interval [0, π/4].
Page 1
Points:
of 10.
Z
3. (6 points) If
7
f 0 (x) dx = 20, f 0 is continuous, and f (1) = 3, find f (7).
1
4. (6 points) Sketch the graph of the function that satisfies the given conditions:
• f (0) = 0 , f is odd
• f 0 (x) < 0 for 0 < x < 2, f 0 (x) > 0 for x > 2,
• f 00 (x) > 0 for 0 < x < 3, f 00 (x) < 0 for x > 3,
• limx→∞ f (x) = −2.
6
5
4
3
2
1
−6 −5 −4 −3 −2 −1
−1
1
2
3
4
5
6
−2
−3
−4
−5
−6
Page 2
Points:
of 12.
5. (8 points) For the differential equation:
f 00 (x) = −9.8
f 0 (0) = 3
f (0) = 1
(a) Find the most general antiderivative of f 00 .
(b) Using the initial data f 0 (0) = 3, find the particular antiderivative.
(c) Find the most general antiderivative of the function in part (b).
(d) Using the initial data f (0) = 1, find the particular antiderivative.
Page 3
Points:
of 8.
6. (7 points) Using The Second Derivative Test specifically, find the x-coordinate of the
local maxima and/or minima of the function (you don’t have to find the y-value)
Z x
g(x) =
2t − 1 dt
−4
7. (8 points) Identify the most general antiderivatives of:
4
(a)
1 + x2
(b)
1 2
x −x+4
2
Page 4
Points:
of 15.
8. (18 points) Compute the integrals:
Z e3
1
(a)
dx
e2 x
Z
sec θ(sec θ − tan θ) dθ
(b)
Z
(c)
√
x
dx
4 − x2
Page 5
Points:
of 18.
9. (6 points) For the function, f (x) = sin x, draw the approximation of
use 4 rectangles and
Rπ
0
sin x dx if we
(a) left endpoint approximation.
1
π/4
π/2
3π/4
π
π/2
3π/4
π
(b) right endpoint approximation.
1
π/4
10. (5 points) Find the most general antiderivative of
Page 6
3 √
− x.
x
Points:
of 11.
Z
11. (6 points) Let g(x) =
1
x
t+1
dt
t4 + 1
0
(a) Find g (x)
(b) Find g 0 (1)
(c) Find g(1)
Page 7
Points:
of 6.