Math 151 WIR 10
Exam 3 Review: Sections 4.3, 4.4, 4.5, 4.6, 4.8, 5.1, 5.2, 5.3, 5.5, 5.7, 6.1, 6.2
1. Find the lim𝑥→1 𝑥 1⁄(1−𝑥) .
2. Differentiate the function 𝑦 = (𝑙𝑛𝑥)𝑡𝑎𝑛⁡(2𝑥) .
3. Differentiate the function 𝑦 = (𝑠𝑖𝑛𝑥)𝑎𝑟𝑐𝑠𝑖𝑛𝑥 .
4. After 500 years, a sample of radium-226 has decayed to 80.4% of its original mass. Find
the half-life of radium-226.
𝑥+1
5. Differentiate the functions 𝑦 = 𝑐𝑜𝑠 −1 (
) and 𝑦 = 𝑡𝑎𝑛−1 ⁡(𝑒 2𝑥 − 𝑙𝑛𝑥).
2𝑥+1
6. Evaluate lim𝑥→0+
𝑙𝑛𝑥
√𝑥
and lim𝑥→0+ (𝑠𝑖𝑛𝑥)𝑥 .
7. Find the absolute maximum and minimum values of the function 𝑓(𝑥) = −√5 − 𝑥 2 in
the interval [−√5, 1].
8. For each of the two functions listed below, find intervals where the function is
increasing/decreasing, local maximum and minimum points, intervals where the function
is concave up/down and inflection points: 𝑓(𝑥) = 𝑒 1/𝑥 and 𝑓(𝑥) = 𝑥 7/3 + 𝑥 4/3 .
9. A rectangle is to be inscribed in a semicircle of radius 2. What is the largest area the
rectangle can have, and what are its dimensions?
10. Find the function ℎ(𝑥) if ℎ′ (𝑥) = 𝑠𝑒𝑐 2 𝑥 +
11. Evaluate the sum⁡⁡∑𝑛
𝑘=1
5
3𝑘−1
1
2√𝑥
and ℎ(𝜋) = 0.
.
12. Set up, but do not evaluate, a four-term Riemann sum to estimate the area between the x-
axis and the graph of 𝑦 = 𝑥 2 + 3 with x between 0 and 2. You may use either left-hand
points, or right-hand points, or midpoints.