Math 115 Practice for Exam 2

Math 115 — Practice for Exam 2

Generated November 8, 2021

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Instructor: Section Number:

1. This exam has 10 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck.

2. Please read the instructions for each individual exercise carefully. One of the skills being tested on this exam is your ability to interpret questions, so instructors will not answer questions about exam problems during the exam.

3. Show an appropriate amount of work (including appropriate explanation) for each exercise so that the graders can see not only the answer but also how you obtained it. Include units in your answers where appropriate.

4. You may not use a calculator. You are allowed one double-sided 8 × 11 inch page of handwritten notes.

5. If you use graphs or tables to obtain an answer, be certain to include an explanation and sketch of the graph, and to write out the entries of the table that you use.

6. You must use the methods learned in this course to solve all problems.

7. You are responsible for reading and following all directions provided for each exam in this course.

Semester Exam Problem Name Points Score

Winter 2016 2 4 10

Winter 2017 2 6 corn snake 9

Winter 2016 2 2 12

Fall 2009 2 7 Kampyle of Eudoxus 14

Winter 2009 2 4 turtle river 16

Winter 2018 2 7 chlorine 10

Winter 2020 2 7 salt shed 9

Fall 2016 2 9 12

Fall 2015 2 4 10

Winter 2014 2 2 triangle 9

Total 111

Recommended time (based on points): 101 minutes

Math 115 / Exam 2 (March 22, 2016) page 5

4 . [10 points] Let h(x) be a twice differentiable function defined for all real numbers x. (So h is differentiable and its derivative h ^′ is also differentiable.)

Some values of h ^′ (x), the derivative of h are given in the table below.

x −8 −6 −4 −2 0 2 4 6 8

h ^′ (x) 3 7 0 −3 −5 −4 0 −2 6

For each of the following, circle all the correct answers.

Circle “none of these” if none of the provided choices are correct.

a . [2 points] Circle all the intervals below in which h(x) must have a critical point.

−8 < x < −6 −6 < x < −2 −2 < x < 2 2 < x < 6 6 < x < 8 none of these

b . [2 points] Circle all the intervals below in which h(x) must have a local extremum (i.e. a local maximum or a local minimum).

−8 < x < −6 −6 < x < −2 −2 < x < 2 2 < x < 6 6 < x < 8 none of these

c . [2 points] Circle all the intervals below in which h(x) must have an inflection point.

−8 < x < −4 −4 < x < 0 0 < x < 4 2 < x < 6 4 < x < 8 none of these

d . [2 points] Circle all the intervals below which must contain a number c such that h ^′′ (c) = 2.

−8 < x < −6 −4 < x < −2 −2 < x < 0 2 < x < 4 6 < x < 8 none of these

e . [2 points] Suppose that h ^′′ (x) < 0 for x < −8, and h(−8) = 7. Circle all the numbers below which could equal the value of h(−10).

−2 −1 0 1 2

none of these

Math 115 / Exam 2 (March 22, 2017) do not write your name on this exam page 7

6 . [9 points] A group of biology students is studying the length L of a newborn corn snake (in cm) as a function of its weight w (in grams). That is, L = G(w). A table of values of G(w) is shown below.

w 5 10 15 20 25

G (w) 24.5 31.6 38.7 44.7 50 G ^′ (w) 2.23 1.58 1.30 1.12 1.05

Assume that G ^′ (w) is a differentiable and decreasing function for 0 < w < 25.

a . [2 points] Find a formula for H(w), the tangent line approximation of G(w) near w = 20.

Answer: H (w) =

b . [1 point] Use the tangent line approximation of G(w) near w = 20 to approximate the length of a corn snake that weighs 22 grams.

Answer:

c . [2 points] Is your answer in part (b) an overestimate or an underestimate? Circle your answer and write a sentence to justify it.

Circle one: Overestimate Underestimate cannot be determined Justification:

d . [4 points] In their study of the growth of corn snakes, they found the results of a recent article that states that the average weight w of a corn snake (in grams) t weeks after being born is given by w = ¹ ₅ t ² . Let S(t) = G ¹ ₅ t ²
be the length of a corn snake t weeks after being born. Find a formula for P (t), the tangent line approximation of S(t) near t = 5.

Answer: P (t) =

University of Michigan Department of Mathematics Winter, 2017 Math 115 Exam 2 Problem 6

Math 115 / Exam 2 (March 22, 2016) page 3

2 . [12 points]

Let f be the piecewise linear function with graph shown below.

−3

−2

−1 1 2 3

−3 −2 −1 1 2 3

x y

y = f (x)

The table below gives several values of a differen- tiable function g and its derivative g ^′ .

Assume that both g(x) and g ^′ (x) are invertible.

x −2 −1 0 2 5

g(x) 21 11 5 −1 −3

g ^′ (x) −12 −8 −4 −2 −0.4

You are not required to show your work on this problem. However, limited partial credit may be awarded based on work shown.

For each of parts a.-f. below, find the value of the given quantity. If there is not enough information provided to find the value, write “not enough info ”. If the value does not exist, write “does not exist ”.

a . [2 points] Let j(x) = e ^g(x) . Find j ^′ (2).

Answer:

b . [2 points] Let k(x) = f (x)f (x + 2). Find k ^′ (−1).

Answer:

c . [2 points] Let h(x) = 3f (x) + g(x). Find h ^′ (−2).

Answer:

d . [2 points] Find (g ⁻¹ ) ^′ (2).

Answer:

e . [2 points] Let m(x) = g(f (g(x))). Find m ^′ (2).

Answer:

f . [2 points] Let ℓ(x) = f(x)

g(2x) . Find ℓ ^′ (−1).

Answer:

Math 115 / Exam 2 (November 24, 2009) page 8

7 . [14 points] The Kampyle of Eudoxus is a family of curves that was studied by the Greek mathematician and astronomer Eudoxus of Cnidus in relation to the classical problem of doubling the cube. This family of curves is given by

a ² x ⁴ = b ⁴ (x ² + y ² ).

where a and b are nonzero constants and (x, y) 6= (0, 0)—i.e.. the origin is not included.

a . [5 points] Find dy

dx for the curve a ² x ⁴ = b ⁴ (x ² + y ² ).

b . [5 points] Find the coordinates of all points on the curve a ² x ⁴ = b ⁴ (x ² + y ² ) at which the tangent line is vertical, or show that there are no such points.

c . [4 points] Show that when a = 1 and b = 2 there are no points on the curve at which the tangent line is horizontal.

University of Michigan Department of Mathematics Fall, 2009 Math 115 Exam 2 Problem 7 (Kampyle of Eudoxus)

5

4. (16 points) The Awkward Turtle is going to a dinner party! Unfortunately, he’s running quite late, so he wants to take the quickest route. The Awkward Turtle lives in a grassy plain (his home is labeled H in the figure below), where his walking speed is a slow but steady 3 meters per hour. The party is taking place southeast of his home, on the bank of a river (denoted by P in the figure). The river flows south at a constant rate of 5 meters per hour, and once he gets to the river, the Awkward Turtle can jump in and float the rest of the way to the party on his back. A typical path the Awkward Turtle might take from his house to the party is indicated in the figure below by a dashed line.

What is the shortest amount of time the entire trip (from home to dinner party) can take? [Recall that rate × time = distance.]

25 m

15 m

T h e R iv er

H

P 3 m/hr

5 m/hr

Minimal time =

Math 115 / Exam 2 (March 20, 2018.) do not write your name on this exam _{page 12}

7 . [10 points] The amount of chlorine in a chemical reaction C(t) (in gallons) t seconds after it has been added into a solution is given by the function

C (t) = 2 − 3(t − 5)

(t − 1)e ^−t for t ≥ 0.

Notice that

C ⁰ (t) = 3(t − 6)(5t − 9)e ^−t 5(t − 5) ^{1 /5} .

a . [8 points] Use calculus to find the time(s) (if any) at which the amount of chlorine in the solution is the greatest and the smallest. If the function has no global maximum or global minimum write “None” in the appropriate space. Show all your work.

Answer: Global maximum(s) at t = Global minimum(s) at t =

b . [2 points] What is the maximum amount of chlorine in the solution? If there is never a maximum amount of chlorine in the solution, write “None”.

Answer:

University of Michigan Department of Mathematics Winter, 2018 Math 115 Exam 2 Problem 7 (chlorine)

Math 115 / Exam 2 (April, 2020) page 8

7 . [9 points]

r r

r

h

A city is in the planning stages of building a shed to store road salt. One design being considered is shown above. The sides would be a cylinder of radius r feet and height h feet, and the roof would be a cone in which both the radius and height are equal to r feet. (The city does not need to build a floor.) The cost of the materials for this shed, in dollars, is

4πr ² + 4πrh.

If the city wants to spend ✩20,000 on materials, what values of r and h will maximize the volume of the shed? Give your answers to at least two decimal places, and be sure to find and justify your answers using calculus.

Note that the volume of a cone with radius R and height H is 1

3 πR ² H .

Math 115 / Exam 2 (November 14, 2016) do not write your name on this exam page 10

9 . [12 points] The graph of a portion of the derivative of a function f (x) is given below.

Assume that the domain of f is all real numbers, and that f is continuous on the entire interval [−2, 5].

−2 −1 1 2 3 4 5

−6

−4

−2 2 4 6

y = f ^′ (x)

x y

Use the graph above to answer the following questions. For each question, circle all of the available correct answers.

(Circle none of these if none of the available choices are correct.)

a . [2 points] At which of the following values of x does f (x) appear to have a critical point?

x = 0 x = 1 x = 2 x = 3 x = 4 none of these

b . [2 points] At which of the following values of x does f ^′ (x) appear to have a critical point?

x = 0 x = 1 x = 3 x = 4 none of these

c . [2 points] At which of the following values of x does f (x) attain a local extremum?

x = −1 x = 0 x = 1 x = 3 none of these

d . [2 points] At which of the following values of x does f (x) attain a global maximum on the interval [−1, 3]?

x = −1 x = 0 x = 1 x = 2 x = 3 none of these

e . [2 points] At which of the following values of x does f (x) have an inflection point?

x = −1 x = 0 x = 1 x = 2 x = 3 none of these

f . [2 points] For which of the following intervals is f (x) concave up on the entire interval?

−1 < x < 0 0 < x < 1 1 < x < 2 2 < x < 4 none of these

University of Michigan Department of Mathematics Fall, 2016 Math 115 Exam 2 Problem 9

Math 115 / Exam 2 (November 17, 2015) page 5

4 . [10 points] A function f (x) is defined and differentiable on the interval 0 < x < 10. In addition, f (x) and f ^′ (x) satisfy all of the following properties:

• f ^′ (x) is continuous on the interval 0 < x < 10.

• f ^′ (1) = 2.

• f ^′ (x) is differentiable on the interval 1 < x < 5.

• f (x) is concave up on the interval 3 < x < 5.

• f (x) has a local minimum at x = 4.

• f (x) is decreasing on the interval 6 < x < 8.

• f (x) has an inflection point at x = 7.

• f ^′ (x) is not differentiable at x = 9.

On the axes provided below, sketch a possible graph of f ^′ (x) (the derivative of f (x)) on the interval 0 < x < 10.

Make sure your sketch is large and unambiguous.

Graph of y = f ^′ (x)

−4

−3

−2

−1 1 2 3 4

x y

1 2 3 4 5 6 7 8 9 10

Math 115 / Exam 2 (March 27, 2014) page 3

2 . [9 points] Consider a right triangle with legs of length x ft and y ft and hypotenuse of length z ft, as in the following picture:

x y

z

a . [2 points] Suppose that the perimeter of the triangle is 8 ft. Let A(x) give the area of the triangle, in ft ² , as a function of the side length x. In the context of this problem, what is the domain of A(x)? Note that you do not need to find a formula for A(x).

Answer:

b . [7 points] Suppose instead that the perimeter of the triangle is allowed to vary, but the area of the triangle is fixed at 3 ft ² . Let P (x) give the perimeter of the triangle, in ft, as a function of the side length x.

(i) In the context of this problem, what is the domain of P (x)?

Answer:

(ii) Find a formula for P (x). The variables y and z should not appear in your answer.

(This is the equation one would use to find the value(s) of x minimizing the perimeter.

You should not do the optimization in this case.)

Answer: P (x) =

University of Michigan Department of Mathematics Winter, 2014 Math 115 Exam 2 Problem 2 (triangle)