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A calculus exam from math 163 section 51 at the university of california, berkeley, held on november 7, 2002. The exam covers topics such as differentiation, limits, and applications of derivatives. It includes various types of questions, some requiring explanations and others being essay questions. The marks earned on these questions depend on the quality of the answers.
Typology: Exams
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Name: Signature:
z − 1 0 1 2 3 4 5 6 m(z) 10. 1 9. 8 8. 1 6. 0 3. 3 0. 6 − 1. 5 − 1. 3 m′(z) − 0. 2 − 0. 9 − 1. 7 − 2. 0 − 2. 6 − 2. 6 − 2. 3 − 1. 1 n(z) 7. 7 3. 2 0. 2 − 1. 4 − 2. 0 − 1. 7 − 1. 0 0. 3 n′(z) − 5. 4 − 3. 7 − 2. 2 − 1. 0 0. 0 0. 2 0. 8 1. 4
(a) Let f (z) = 3m(z) −
n(z) 2
. Find f ′(1).
f ′(1) = 3m′(1) −
n′(1) 2
(b) Let g(z) = m(n(z)). Find g′(5).
g′(5) = m′(n(5))n′(5) = m′(− 1 .0)n′(5) = (− 0 .2)(0.8) = − 0. 16
(c) Let h(z) = m(z)/n(z). Find h′(0).
h′(0) =
m′(0)n(0) − m(0)n′(0) (n(0))^2
Let V denote the volume of the Dandelion Creek tower (in cubic feet), r the radius of the base (in feet), and h the height of the tower (in feet). We can then write
V = πr^2 h.
Let t indicate time in days. Then differentiating everything in sight with respect to t, we get dV dt
= 2πrh
dr dt
dh dt
Since the volume of the tower stays constant, dV /dt = 0. We also know that dh/dt = − 3. Thus, we know the values of all of the quantities in the above equation except for dr/dt. We hence have that
0 = 2π(8)(50)
dr dt
− π(8)^2 (−3)
or 0 = 800π
dr dt
− 192 π.
Solving for dr/dt, we get
dr dt
= 0. 24 feet per day.
(a) Sketch the trajectory of the particle on the xy-plane.
x = f (t) y = g(t)
(b) Estimate the speed of the particle at t = 6.
The speed is (^) √ (f ′(6))^2 + (g′(6))^2 ≈
Note that we estimate f ′(6) and g′(6) graphically by examining the slopes of the appropriate tangent lines.
Solutions may vary. However, we can ascertain the following properties of P :
Graph of P ′
Good solutions should...
... describe properties of P which follow from the given criteria ... clearly explain why the properties of P must follow from the above criteria. ... be well-organized, with few grammatical and spelling errors. ... utilize a combination of graphical, numerical, and algebraic methods in the explanation.