MATH 147, SPRING 2016
COMMON EXAM I (PART 2) - VERSION A
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FIRST NAME:
INSTRUCTOR:
SECTION NUMBER:
UIN:
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
8
4
8
5
14
Part 1
50
100
1
1. Radium-226 (Ra226 ) has a half-life of 1590 years.
(a) (5 pts) If a sample has an initial mass of 100 mg, find a formula for the mass remaining after t years.
(b) (5 pts) When will the mass be reduced to 20 mg? (Express your answer in terms of the natural logarithm and
include the unit of time).
2
2. (10 pts) Use the double log plot below to determine the functional relationship between x and y.
10 5
10 4
y
10 3
10 2
10 1
10 0
10 0
10 1
10 2
x
3
3. Let f be the function defined by


cx2 + 2x,
if x < 2
8,
if x = 2
f (x) =
 3
x + cx − 3, if x > 2
where c is a fixed constant.
(a) (2 pts) Find lim− f (x) and lim+ f (x) in terms of c.
x→2
x→2
(b) (4 pts) Find the value of c such that lim f (x) exists.
x→2
(c) (2 pts) For the value of c you found above, is f continuous at x = 2? Explain your answer.
4
4. (8 pts) Show that there is a solution of the equation x5 + x − 3 = 0 in the interval (1, 2).
State the name(s) of any theorem(s) that you use in your argument.
5
5. Consider the function f (x) =
√
x − 5.
(a) (10 pts) Find f 0 (x) using the limit definition of the derivative.
(b) (4 pts) Find an equation of the tangent line to the graph of y =
Express your answer in slope-intercept form.
6
√
x − 5 at x = 9.