MATH 150 TEST 2
SPRING, 2011
Remember to keep your work neat and orderly. Show all of your work. NO WORK = NO CREDIT! Read each
question carefully and be sure to answer the question that was asked. You must use correct notation. Calculators
may be used to check answers, but not to give answers. Good luck!
Name:____________________________________________________________
pts
1. Find the absolute maximum and absolute minimum function values on the interval [-10, 1] for
𝑓(𝑥) = 5 − 6𝑥 − 𝑥 2 .
8
Max = 14; min = -35
2. Find
𝑑𝑦
𝑑𝑥
for the implicitly defined function
𝑥
3
− 4𝑥 2 𝑦 3 = 3 csc 2𝑦. Solve for
𝑑𝑦
𝑑𝑥
!
8
1
− 8𝑥𝑦 3
𝑑𝑦
3
=
𝑑𝑥 12𝑥 2 𝑦 2 − 6 csc 2𝑦 cot 2𝑦
3. Find and simplify derivatives of the following functions (no negative exponents!):
3
a. 𝑊 = 12√𝑧 2 − 2𝑧 6 ln 𝑧 + 5 tan 3𝑧
𝑊′ =
b. 𝑃 = 2 sin4 𝑡 2 −
1
4𝑡
8
𝑧 1/3
4
9
− 12𝑧 5 ln 𝑧 − 2𝑧 5 + 15 sec 2 3𝑧
+ 2𝑒 1−𝑡 +
3
1
2(5−3𝑡)6
𝑃′ = 16𝑡 sin3 𝑡 2 cos 𝑡 2 +
3
9
3 1−𝑡 4
−
8𝑡
𝑒
+
(5 − 3𝑡)7
4𝑡 4
12
4. If the velocity at time t of an object moving in a straight line (in ft/sec) is given by
1
𝑣 = 6 + 𝑒 3−3𝑡 , find the acceleration (including units!) at t = 1 sec.
2𝑡
𝑎 = −6
6
𝑓𝑡
𝑠𝑒𝑐 2
5. If the surface area (𝑆 = 4𝜋𝑟 2 ) of a spherical balloon is decreasing at a constant rate of
400 𝑐𝑚2 /𝑠𝑒𝑐, find the rate at which the radius is changing (including units!) when the
radius is 5 cm.
7
−10 𝑐𝑚
𝜋 𝑠𝑒𝑐
6. Use the quotient rule to find the derivative (do not simplify!) of
𝐺′ =
(3 +
𝐺=
1
𝑢
(1+𝑢2 )5
3𝑢−
7
1
1
) (1 + 𝑢2 )5 − (3𝑢 − 𝑢) 5(1 + 𝑢2 )4 2𝑢
2
𝑢
(1 + 𝑢2 )10
7. Find all critical points for the function
𝑓(𝑥) = 4(𝑥 2 + 7𝑥 + 10)2/3
7
7
𝑥 = − , −5, −2
2
8. If 𝑧 = 2 ln(𝑦 2 + 3𝑦 + 7) and 𝑃 = (2 − 3𝑧)7 , use the chain rule (two different derivatives) with
correct notation to find the rate of change of P with respect to y as a function of y only. Do not
simplify!
7
𝑑𝑃 𝑑𝑃 𝑑𝑧
2(2𝑦 + 3)
=
= 7(2 − 3 (2)ln(𝑦 2 + 3𝑦 + 7))6 (−3) 2
𝑑𝑦 𝑑𝑧 𝑑𝑦
𝑦 + 3𝑦 + 7
9. If the derivative of a function is given by 𝑓 ′ = −3(5 + 𝑥)4 (2 − 𝑥)(1 + 𝑥),
a. Find all open intervals where the function is increasing and decreasing. Fully justify!
6
(−∞, −5) 𝑖𝑛𝑐
(−5, −1) 𝑖𝑛𝑐
(1, 2)
𝑑𝑒𝑐
(2, ∞) 𝑖𝑛𝑐
b. Identify each critical point as a relative min, max, or neither.
X = -5 neither; x = -1 rel max; x = 2 rel min
10.Find one set of specific constant values for a, b, and c such that the function
𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 has a critical point at x = 3.
3
5
𝑎 = 1, 𝑏 = −6, 𝑐 = 0
𝑑
−1
𝑑𝑢
11.Given that (cos −1 𝑢) =
, find the derivative of
𝑑𝑥
√1−𝑢2 𝑑𝑥
simplify.
𝑃′ = 2
𝑃 = 2 cos −1 (3𝑧 + 5). Do not
6
−3
√1 − (3𝑧 + 5)2
12.Find the third derivative of the function 𝑦 = 2 ln 𝑥 +
𝑦 ′′′ =
𝑒 4𝑥
4
+ 32𝑒 4𝑥
3
𝑥
2
− √7 (no negative exponents!)
9