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Answer Key 1
MATH 10560: Calculus II Practice Exam 1.1 September 25, 2007
Record your answers to the multiple choice problems by placing an × through one letter
for each problem on this page. There are 8 multiple choice questions worth 6 points each
and 4 partial credit problems worth 10 points each. You start with 12 points. On the
partial credit problems try to simplify your answer and indicate your final answer clearly.
You must show your work and all important steps to receive credit.
You may not use a calculator.
- a b c d •
- • b c d e
- a b c d •
- a b c d •
- a b • d e
- a b c • e
- a b c • e
- a b c • e
MATH 10560: Calculus II Practice Exam 1.1 September 25, 2007
Record your answers to the multiple choice problems by placing an × through one letter
for each problem on this page. There are 8 multiple choice questions worth 6 points each
and 4 partial credit problems worth 10 points each. You start with 12 points. On the
partial credit problems try to simplify your answer and indicate your final answer clearly.
You must show your work and all important steps to receive credit.
You may not use a calculator.
- a b c d e
- a b c d e
- a b c d e
- a b c d e
- a b c d e
- a b c d e
- a b c d e
- a b c d e
- Evaluate
∫ (^1) /√ 2
0
sin − 1 (x) √ 1 − x^2
dx.
(a) − ln(
- (b) π/ 4 (c) 1 − π 2 / 16 (d) π 2 / 32 (e)
- Use L’Hospital’s Rule to evaluate lim x→π
sin(x)
ln(π/x)
(a) 0 (b) 1 (c) π 2 (d) π (e) ∞
- Use a trigonometric substitution to evaluate
∫ (^2)
1
x^2
4 x^2 − 1
dx.
(a)
(b) 3
(c)
3 (d)
3 (e)
- Let f (x) =
1 − e^2 x
1 + e^2 x^
a) Find the domain and range of f (x).
b) Show that f (x) is one-to-one in its domain.
c) Find f − 1 (x) and determine its domain and range.
- A photographer is taking a picture of a three-foot tall painting hung on a wall in an art
gallery. The camera is one foot below the lower edge of the painting, as shown in the figure below. Determine the distance x the camera should be from the wall to maximize the angle θ subtended by the camera lens. Hint: Let φ be the angle to the bottom of the picture as shown in the figure. Find the tangent of θ + φ and the tangent of φ, then use inverse functions to solve for θ as a function of x. Be sure to explain why your answer maximizes θ and does not minimize it.
θ
3
1 φ x
- Compute the limit lim x→∞
( 1 +
x − 1
) 3 x .
- Use integration by parts to evaluate
∫ (^3)
0
x 2 e x/ 3 dx.
