Math 151 WIR 8: Sections 4.3, 4.4, 4.5, 4.6, 4.8
1. Find the inverse function of each of the following:
𝑦 = (log 5 𝑥)2
a.
b. 𝑦 =
1+𝑒 𝑥
c. 𝑦 = 23
1−𝑒 𝑥
𝑥
2. Solve each of the following for x.
𝑙𝑛 (
a.
𝑥−3
𝑥+2
) = 3 + 𝑙𝑛 (
𝑥−1
𝑥+2
b. 7𝑒 𝑥 − 𝑒 2𝑥 = 12
)
3. Evaluate the limits:
𝑥
a.
lim𝑥→0+ ln⁡(𝑡𝑎𝑛𝑥)
c. lim𝑥→−∞ 𝑒 3 𝑐𝑜𝑠𝑥
b.
lim𝑥→0 (1 + 𝑥)2/𝑥
d. lim𝑥→−∞ ln (𝑥 4 )
4. Differentiate each of the following functions.
𝑓(𝑥) = 𝑙𝑛 (
a.
𝑎−𝑥
𝑎+𝑥
b. 𝑔(𝑦) = ln⁡(𝑦 3 𝑠𝑖𝑛𝑦)
)
c. ℎ(𝑡) = 𝜋 −𝑡
𝑡𝑎𝑛−1 𝑥
d. 𝑔(𝑥) = 𝑥 1.6 + 1.6𝑥
e. ℎ(𝑥) = (𝑥 √𝑥)
5. Find ℎ′(𝑒), where h is the inverse of 𝑓(𝑥) = 𝑒 𝑥 + 𝑙𝑛𝑥.
𝑑𝑦
6. Find 𝑑𝑥 if 𝑥𝑦 = ln⁡(𝑥 2 + 𝑦 2 ).
7. Evaluate the limits below.
lim𝑥→−∞ 𝑡𝑎𝑛−1 (−𝑥 4 )
a.
b. lim𝑥→−1 𝑐𝑜𝑠 −1 (
𝑥
𝑥+1
c. lim𝑥→∞ 𝑠𝑖𝑛 −1 (
)
8. Find the derivatives of the following functions.
5
𝑦 = 𝑐𝑜𝑠 −1 (𝑡𝑎𝑛−1 𝑥)
a.
b. 𝑦 = 𝑠𝑖𝑛−1 (5𝑡 )
c. 𝑦 =
(𝑐𝑜𝑠 3 𝑥)( √𝑥 2 −1)
(𝑥 4 +1)7 (𝑥 2 +3)8
9. Compute the following:
23
a.
𝑠𝑖𝑛−1 (𝑠𝑖𝑛 (−
c.
𝑎𝑟𝑐𝑐𝑜𝑠 (𝑐𝑜𝑠 ( 𝜋))
e.
𝑠𝑖𝑛−1 (−
g.
𝑡𝑎𝑛−1 (−1)
12
𝜋))
5
13
b.
𝑡𝑎𝑛−1 (𝑡𝑎𝑛 ( 𝜋))
d.
𝑐𝑜𝑠(𝑐𝑜𝑠 −1 (0.37))
f.
𝑐𝑜𝑠 −1 (−
h.
𝑐𝑜𝑠 −1 (0)
3
4
√3
2
)
3
10. Evaluate cos⁡(2𝑠𝑖𝑛−1 (− 5)). Also evaluate sec⁡(𝑎𝑟𝑐𝑡𝑎𝑛2).
√3
2
)
.
𝑒𝑥
𝑒 2𝑥 +𝑒 −𝑥
)
11. Find the following limits:
a.
lim𝑥→−∞ 𝑥𝑒 𝑥
b.
𝜋
lim𝑥→𝜋− (𝑥 − ) 𝑡𝑎𝑛𝑥
2
2
d.
lim𝑥→+∞
(𝑒 𝑥 )0.002
𝑥
e.
lim𝑥→+∞ (
𝑥+1 2𝑥−1
𝑥−2
)
c.
lim𝑥→+∞
f.
lim𝑥→1 (
(ln⁡𝑥)100
𝑥
1
𝑙𝑛𝑥
−
1
𝑥−1
)
12. Thanks to S. F. Ellermeyer This problem will show you how you are able to know the temperature of the inside of your
refrigerator without placing a thermometer in the refrigerator. Take a can of soda out of the refrigerator and leave it in the
room for half an hour, then record the temperature. Leave it for another half hour and record its temperature again. Suppose
that you recorded 𝑇(1/2) = 45° 𝐹 and 𝑇(1) = 55° 𝐹. If the room’s temperature is 70° 𝐹, what is the temperature inside the
refrigerator?