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Math 8: Practice Test 1
Please write neatly and clearly. On all problems, you must show your work and identify
your final answer. Partial credit will be given. You may use a calculator. However, no cell phones or
books may be used.
1. Calculate the derivative of the following functions.
r
x2 + 4
a. f (x) = ln
x2 − 4
b. f (x) = x2 e3x
c. y = arcsin(4x + 2)
2. Calculate the following anti-derivatives.
Z
2
a.
xe−3x dx
Z
4
b.
dx
3x − 1
Z
x+3
c.
dx
x−1
Z
dx
d.
9 + 4x2
3. Find the critical points and points of inflection of y = e−x
2
/b
, where b > 0 is a constant.
4. In 2009, the population of Mexico as 111 million and growing 1.13% annually, while the population of
the US was 307 million and growing .975% annually. If we measure growth rates in people/year, which
population was growing faster in 2009?
5. A yam is put in a hot oven, maintained at a constant temperature of 200◦ C. At time t = 30 minutes, the
temperature T of the yam is 120◦ and is increasing at an (instantaneous) rate of 2◦ /min. Newton’s law
of cooling (or, in our case, warming) implies that the temperature T at time t satisfies the differential
equation
dT
= −k(T − 200).
dt
a. Determine the temperature function T (t).
b. When will the yam reach a temperature of 135◦ ?
6. Let f (x) = log2 (x).
a. Calculate f (5).
b. Use the change of base formula to rewrite f (x) in terms of ln.
c. Calculate the derivative f 0 (x).