MATH 150 - SP 2016
Final Exam – May 11, 2016
Name: _________________________________________________________ Score:
Dr. Okkyung Cho
/200 =
%
 Show neat and organized work! Do not use a calculator unless stated otherwise.
1. [20 points] Consider the following graph of the function 𝑓 . Answer for questions below.
For (g) – (j), only explanation earns credit.
(a)
lim 𝑓(𝑥) =
𝑥→−1−
(b) 𝑓(−1) =
(c)
lim 𝑓(𝑥) =
𝑥→2
(d) lim 𝑓(𝑥) =
𝑥→4
(e) lim 𝑓(𝑥) =
𝑥→0
(f) lim 𝑓(𝑥) =
𝑥→7
(g) Is 𝑓 continuous at 𝑥 = 4? Explain using the limit definition of continuity.
(h) Find any 𝑥 −value(s) at which 𝑓 is continuous but not differentiable. Explain why.
(i) Determine if 𝑓′(1) is positive, negative, or zero. Explain why.
(j) Determine if 𝑓′′(5) is positive, negative, or zero. Explain why.
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2. [40 points] Find the derivatives using various derivative rules. Simplify and leave your answers
without negative exponents.
(a) 𝑓(𝑥) = 4𝑥 5 − 5𝑥
(𝑏) 𝑔(𝑥) =
2
+ 14
𝑥3
(c) ℎ(𝑥) = 𝑒 𝑥 ln 𝑥
(d) 𝑓(𝑥) =
1
1 − 2𝑥
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2. continued
(𝑒) 𝑔(𝑥) = 2𝑒 3𝑥−4
(𝑓) ℎ(𝑥) =
ln(𝑥 5 )
5
(g) 𝑓(𝑥) = √𝑥 2 + 3
(h) 𝑔(𝑥) = ln(3𝑥 4 + 2𝑥)
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3. [20 points]The graph at right is the derivative of a function 𝑓. Show work clearly.
(a) Use the Increasing/Decreasing Test to
determine where 𝑓 is decreasing or increasing.
(b) At what values of 𝑥 do local maxima occur?
Minima?
(c) Determine where 𝑓 is concave up or concave
down.
(d) Determine the 𝑥 −coordinates of the points of
inflection.
(e) Draw a graph of the function 𝑓(𝑥). Note that
you are not expected to determine any
𝑦 −values.
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4. [20 points] The population of a city has a constant relative growth rate. You want to use
continuous exponential growth model to describe the population 𝑃(𝑡) over time, where 𝑡 is years
since 1990.
(a) [8 points] Give the model if the initial population in 1990 was 350,000 and by the year 2000
the population has increased to 420,000. Round your growth rate to 4 decimal places.
(b) [4 points] Use your model to predict the population in 2010.
(c) [8 points] Determine 𝑃′ (20), rounding to an appropriate unit. Then interpret your result using
appropriate units.
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5. [20 points] A white-water rafting company knows that at a price of $80 per person for a certain
trip, they will attract 300 customers. When the price falls to $79, they attract 306 customers.
(a) [8 points] Use this information to write and simplify a linear demand function for this
situation. Write demand (price) as a function of quantity 𝑞.
(b) [4 points] Determine the revenue function for this company.
(c) [8 points] Determine the price the company should charge in order to maximize revenue. Use
the calculus to verify that the price maximizes revenue and does not minimize revenue.
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6. [16 points] Find one antiderivative for each of the following functions, simplifying your answer
and eliminating negative exponents.
(𝑎) 𝑦 ′ = 3√𝑥
(𝑐) 𝑦 ′ =
𝑥 3
+
3 𝑥
(𝑏) 𝑦 ′ = 𝑒 7𝑥−16
(𝑑) 𝑦 ′ = 6𝑥 4 + 15
7. [12 points] Evaluate the following definite integrals.
3
(𝑎) ∫
1
3
𝑑𝑥
𝑥2
4
(𝑏) ∫ 𝑒 −𝑥 𝑑𝑥
0
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8. [14 points] (a) Use a Riemann sum with 5
subintervals and right endpoints to estimate the
5
integral ∫−5 𝑓(𝑥) 𝑑𝑥 for the function 𝑓(𝑥) in the
given graph. Show your work in the space below
and on the diagram at right.
5
(b) What is the actual value of ∫−5 𝑓(𝑥) 𝑑𝑥?
20
9. [10 points] Consider the function 𝑓(𝑥) = 3𝑥+4. Find an equation of the line tangent to 𝑓(𝑥) at
the point (−2, −10). Put your equation into slope-intercept form.
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10. [20 points] Follow the steps below in order to find the area of the region bounded by
𝑓(𝑥) = 2 − 𝑥 and 𝑔(𝑥) = 4 − 𝑥 2 . Make sure you actually show all of the steps, each step earns
points.
(a) [5 points] Find the points of intersection of the two graphs. State the coordinates of these
points.
(b) [5 points] Sketch the graphs of the two functions on a coordinate system below, and label each
function and the points of intersection. Shade the region bounded by the two functions.
(c) [5 points] Write the integral that represents the area of the shaded region and simplify the
integral.
(d) [5 points] Use calculus to find the area of the shaded region in part (b).
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11. [8 points ] Given the function 𝑓(𝑥, 𝑦) = 9𝑥𝑦 + 3𝑒 𝑥 𝑦 2 + 4𝑥 − 7𝑦, determine the following
partial derivatives:
(𝑎) 𝑓𝑥 (𝑥, 𝑦)
(b) 𝑓𝑦 (𝑥, 𝑦)
(𝑐) 𝑓𝑥𝑥 (𝑥, 𝑦)
(d) 𝑓𝑦𝑥 (𝑥, 𝑦)
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