MATH 148, SPRING 2016
COMMON EXAM I (PART 2) - VERSION A
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FIRST NAME:
INSTRUCTOR:
SECTION NUMBER:
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DIRECTIONS:
1. The use of a calculator, laptop or computer is prohibited.
2. Present your solutions in the space provided. Show all your work neatly and concisely and clearly indicate your
final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work
leading up to it.
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“An Aggie does not lie, cheat or steal, or tolerate those who do.”
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Question
Points Awarded
Points
1
10
2
10
3
10
4
10
5
10
Part 1
50
100
1
Z
1. (10 pts) Evaluate
2x3 − x2 + 2
dx.
x4 + x2
2
2. (10 pts) Solve the differential equation
dy
2xy
= 2
dx
x +4
with initial condition y(0) = 8.
3
Z
3. (10 pts) Use a substitution to evaluate
0
1
(x2
6x + 12
dx.
+ 4x + 1)2
4
Z
4. (10 pts) Evaluate
x5 cos(x2 ) dx.
(Hint: First make a w-substitution.)
5
5. Consider the function f (x) = ln x.
(a) (5 pts) Find the third-degree Taylor polynomial, T3 (x), for f (x) at a = 2.
(b) (5 pts) Find the maximum possible error of the approximation f (x) ≈ T3 (x) for x ∈ [1, 3].
Taylor’s Inequality: If |f (n+1) (x)| ≤ M , then
|Rn (x)| ≤
6
M
|x − a|n+1 .
(n + 1)!