math 251 final exam review(answers) - university of...

Math 251 Final Exam Review (Answers) Fall 2017

Below are a set of review problems that are, in general, at least as hard as the problemsyou will see on the final exam. You should know the formula for area of a circle, square, andtriangle. All other area formulae needed on the exam will be given to you.

1. Find the indicated limit.

(a) limx→1+

ex2 − 1

1− x= −∞

(b) limx→π−

ln(sin(x)) = −∞

(c) limt→2

t2 − 4

t3 − 8=

1

3

(d) limv→4+

4− v|4− v|

= −1

(e) limx→3

√x+ 6− xx3 − 3x2

=−5

54

(f) limx→−∞

4− 3x3 − x4

5 + x− 7x4=

1

7

(g) limx→∞

1 + 2x3

3− x− 7x4= 0

(h) limx→1−

(1

x− 1− 1

x2 − 3x+ 2

)= −∞

(i) limx→∞

ex2−x=∞

(j) limx→0+

(1 + sin(x))cot(x)= e

2. Let g(t) = |t− 2|.

(a) For what values of t is g(t) continuous?g(t) is continuous for all real values of t, i.e. (−∞,∞).

(b) For what values of t is g(t) differentiable?(−∞, 2) ∪ (2,∞).

3. Consider the graph of y = f(x), shown below on the domain [−5, 5].

[Recall that • represents a point that is defined on the graph, while ◦ represents apoint that is not defined on the graph.]

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x-4 -2 2 4

y

-4

-2

2

4

(a) Compute each of the following limits, if possible. If a limit does not exist, write“DNE”.

i. limx→−2

f(x) DNE

ii. limx→0−

f(x)= 0

iii. limx→1

f(x) = 1

iv. limx→−1

f(x) = 2

(b) Is f continuous at x = −1? Use limits to justify your conclusion.No, since f(−1) = −2 and hence

limx→−1

f(x) 6= f(−1)

(c) For which value(s) of x is f not differentiable?x = −2,−1, 0

4. Find the indicated derivative.

(a) f ′(x), where f(x) = (x4 − 2x2 + 3)3

f ′(x) = 3(x4 − 2x2 + 3)2(4x3 − 4x)

(b) g′(t), where g(t) =√t+

t− 1

ln(t)

g′(t) =1

2√t

+ln(t) + t−1 − 1

(ln(t))2

(c) y′′, where y = esin(3θ)

y′′ = 9esin(3θ) cos2(3θ)− 9esin(3θ) sin(3θ)

2


(d) h′(θ), where h(θ) = cos2(tan(θ))h′(θ) = −2 sec2(θ) sin(tan(θ)) cos(tan(θ))

(e) f ′(x), where f(x) = e2x ln(tan(x))

f ′(x) = 2 ln(tan(x))e2x + e2x · sec2(x)

tan(x)

(f)dy

dx, where xy4 + x2y = x+ 3y

dy

dx=

1− y4 − 2xy

x2 + 4xy3 − 3

(g)dx

dt, where t2 cos(x) + sin(3t) = tx

dx

dt=

2t cos(x) + 3 cos(3t)− xt2 sin(x) + t

5. Find an equation of the tangent line to the curve at the given value of x.

(a) y = (2 + x)e−x; x = 0y = 2− x

(b) y =x2 − 1

x2 + 1; x = 0

y = −1

(c) x3y − 5xy = 4; x = −1

y =1

2(x+ 1) + 1

6. At what value(s) of x is the tangent line to the curve f(x) = 4+3ex(1+x2) horizontal?

At x = −1.

7. For each function below:

i. Find all intercepts.

ii. Find the vertical and horizontal asymptotes, if any.

iii. Find the intervals of increase or decrease.

iv. Find the local maximum and minimum values.

v. Find the intervals of concavity and the inflection points.

Then use this information to sketch a graph of the function.

(a) f(x) = x4 + 4x3

i. (0, 0) and (−4, 0)

ii. None

iii. Increase: (−3, 0) ∪ (0,∞); Decrease: (−∞,−3)

iv. f(x) has a local minimum at (−3,−27); f(x) has no local maxima.

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v. f(x) is concave up on (−∞,−2) ∪ (0,∞); f(x) is concave down on (−2, 0).f(x) has inflection points at (−2,−16) and (0, 0).

You can use technology to view the graph of this function in order to check youranswer.

(b) f(x) =x

x2 − 4

i. (0, 0)

ii. f(x) has vertical asymptotes at x = −2, x = 2; f(x) has a horizontal asymp-tote at y = 0.

iii. Decrease: (−∞,−2) ∪ (−2, 2) ∪ (2,∞); f(x) is never increasing

iv. None

v. f(x) is concave up on (−2, 0) ∪ (2,∞); f(x) is concave down on (−∞,−2) ∪(0, 2). f(x) has an inflection point at (0, 0).

You can use technology to view the graph of this function in order to check youranswer.

8. Find all local extrema of the function. Then find the absolute extrema of the functionon the given interval. [You should include both where the extrema occur, as well asthe values of the extrema.]

(a) f(x) = x√

1− x on [−1, 1]

The function has a local maximum at

(2

3,

2

3√

3

). The function does not have

any local minima. The absolute maximum on [−1, 1] is2

3√

3and occurs when

x =2

3. The absolute minimum on [−1, 1] is −

√2 and occurs when x = −1.

(b) f(x) =ln(x)

xon [1, 3]


(e,

1

e

). The function does not have any

local minima. The absolute maximum on [1, 3] is1

eand occurs when x = e. The

absolute minimum on [1, 3] is 0 and occurs when x = 1.

(c) f(x) =3x− 4

x2 + 1on [−2, 2]


(3,

1

2

). The function has a local minimum

at

(−1

3,−9

2

). The absolute maximum on [−2, 2] is

2

5and occurs when x = 2.

The absolute minimum on [−2, 2] is−9

2and occurs when x =

−1

3.

9. Dan, the owner of the Dan’s Pizza Pi company, is reviewing his daily sales figures andpricing structure for his specialty pizza. He has found that in order to sell x pizzas he

4


needs to charge p(x) dollars for each pizza where p(x) = 40e−0.01x for values of x in theinterval [0, 150].

(a) Find the function R(x) that gives Dan’s revenue from the sales of his specialtypizza. [Hint: Revenue is price multiplied by quantity sold.]R(x) = 40xe−0.01x

(b) It is in Dan’s best interest to maximize the amount of revenue he makes each day.How many pizzas must Dan sell to maximize revenue?100 pizzas

(c) What price does Dan need to charge for each pizza in order to sell the number ofpizzas that will maximize his revenue?About $14.72

10. Adam and Justin leave a cafe at 2pm. Adam walks directly west at a rate of 3 milesper hour. Justin walks directly south at a rate of 4 miles per hour.

(a) How fast is the distance between Adam and Justin changing at 3pm?5 miles per hour

(b) Gavin is standing 0.75 miles east of the cafe making a phone call. From whereGavin is standing, he can measure the angle between the street and Justin, asJustin walks south. How fast is this angle changing at 2:15pm?48

25radians per hour

(c) At 2pm, Deb is 5 miles north of the cafe and begins to ride her bike. She ridesdirectly east at a pace of 9 miles per hour. How is the distance between Adamand Deb changing at 3pm?144

13miles per hour

11. The volume of a right circular cone is1

3πr2h, where r is the radius of the base and h

is the height.

(a) Find the rate of change of the volume with respect to the height if the radius isconstant.dV

dh=

1

3πr2

(b) Find the rate of change of the volume with respect to the radius if the height isconstant.dV

dr=

2

3πrh

(c) Find the rate of change of the volume with respect to time, when the radius is 3m and the height is 4 m. Assume the rate of change happens equally for r and hwith a magnitude of 2 m/s.22π m3/s

12. A conical water tank with vertex down has a radius of 4 m at the top and is 16 m high.Assume water is being pumped into the tank at a rate of 2 m3/min.

5


(a) Find the rate at which the water level is rising when the water is 3 m deep.32

9πm/min

(b) Find the rate at which the water level is rising after2

3π minutes.

2

πm/min

(c) What is the depth of the water at the moment when the water level is rising at arate of 4 m/min?2√

2√pi

m

13. A particle is moving along the curve y =√x+ 1. As the particle passes through the

point (3, 2), its x-coordinate increases at a rate of 2 cm/s. How fast is the distancefrom the particle to the origin changing at this instant?

7√13

cm/s

14. A demographic study of a certain city indicates that P (r) hundred people live r milesfrom the civic center, where

P (r) =5(3r + 1)

r2 + r + 2

(a) For what values of r is P (r) increasing? For what values is it decreasing?

P (r) is increasing on

(−5

3, 1

). P (x) is decreasing on

(−∞, −5

3

)and (1,∞).

(b) At what distance from the civic center is the population largest? What is thislargest population?The population is largest 1 mile from the civic center, and this largest populationis 500 people.

15. A 5-year projection of population trends suggests that t years from now, the populationof a certain community will be P (t) = −t3 + 9t2 + 48t + 50 thousand. At what timeduring the 5-year period will the population be growing most rapidly?When t = 3.

16. Find two positive real numbers whose product is 7 and whose sum is as small aspossible.x =√

7, y =√

7

17. A cylindrical can (with a top and bottom) is to be made with 600 π cm2 of aluminum.Find the dimensions (in cm) that will maximize the volume of the can. [The volumeof a cylinder is πr2h and the surface area of the sides of a cylinder equals 2πrh.]r = 10 cm, h = 20 cm

18. Find the shortest distance from the point (5, 0) to the curve 2y2 = x3.√13 (when x = 2).

6


19. If 4800 cm2 of material is available to make a box with a square base and an open top,find the largest possible volume of the box.32000 cm3

20. A manufacturing company is asked to create special bricks for a new retaining wall tobe built in Eugene. The bricks are to be rectangular with the length equal to threetimes the width. The top and bottom of the bricks is to be coated with a reinforcedmaterial that costs 10 cents per in2. The other sides of the bricks will be coated witha standard coating that costs 6 cents per in2. Find the dimensions of the bricks thatwill minimize the cost of production if the bricks must have volume 50 in3.

w = 3

√20

3≈ 1.8821 cm, l = 3 · 3

√20

3≈ 5.6463 cm, h ≈ 4.7050 cm

21. Use the function f(x) = 3√

1 + 3x at a = 0 to estimate the value of 3√

1.03 by using alinear approximation formula.3√

1.03 ≈ 1.01

22. Use the function f(x) =√x− 1 to estimate the value of

√4.4 using a linear approxi-

mation formula.√4.4 ≈ 2.1

23. Give an approximation for ln(0.9) using a linear approximation formula.ln(0.9) ≈ −0.1

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math 251 final exam review(answers) - university of...

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