Math 1110

Prelim 3

21 April 2009

Name:

Instructor:

Section:

INSTRUCTIONS — PLEASE READ THIS NOW• This test has 6 problems on 14 pages, worth a total of100 points. Please carefully write all your final answerson the page they are posed. There are extra blank pagesat the back of this booklet, but we will not read your workon those pages.• Write your name, your instructor’s name, and your sec-tion number right now.• Look over your test packet as soon as the exam begins!If you find any missing pages or problems please ask aproctor for another test booklet.• SHOW YOUR WORK. To receive full credit, your an-swers must be neatly written and logically organized. Ifyou need more space, write on the back side of the pre-ceding sheet, but be sure to label your work clearly. Youdo not need to simplify your answers unless explicitlyinstructed to do so.• You have 11

2hours to complete this exam.

• This is a closed book exam. You are NOT allowed to usea calculator. You may bring a single 3× 5 inch index cardof notes. All other aids are prohibited.• Academic integrity is expected of all Cornell Universitystudents at all times, whether in the presence or absenceof members of the faculty. Understanding this, I declareI shall not give, use, or receive unauthorized aid in thisexamination.

Please sign below to indicate that you have read andagree to these instructions.

Signature of Student

OFFICIAL USE ONLY

1. / 16

2. / 20

3. / 20

4. / 9

5. / 15

6. / 20

Total: / 100

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 1

Question 1. (16 points) Short answer.

(a) (4 points) Please evaluate limx→1

xa − 1

xb − 1, where a and b are positive numbers.

(b) (4 points) Please evaluate limx→π

sin(x)

1 − cos(x).

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 2

(c) (4 points) Please evaluate limx→−∞ x2ex.

(d) (4 points) Please write the sum

7 + 1 + 7 + 2 + 7 + 3 + 7 + 4 + 7 + 5 + · · ·+ 7 + (n − 1) + 7 + n

in “sigma notation” and find a closed form expression for it. What is the value ofthe sum when n = 10?

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 3

Question 2. (20 points) True/False. Determine whether the following statements are trueor false, and circle your response. Please give a brief explanation (in a complete sentence!).

(a) (4 points) If a differentiable function f is defined at c, and c is a critical point of f,then f has either a local minimum or a local maximum at c. TRUE FALSE

(b) (4 points) The function f(x) = sin(cos(x)) +x3 + 7

exis integrable on the interval [a, b]

for any real numbers a and b. TRUE FALSE

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 4

(c) (4 points) Let f(x) be a function, and Pn(x) its order n Maclaurin polynomial. Sup-pose that Pn(0.1) = 3.728 and |En(0.1)| ≤ 0.005. Then we know for certain thatf(0.1), rounded to two decimal places, is 3.73. TRUE FALSE

(d) (4 points) The area of the region bounded by sin x and the x-axis, between x = −π

and x = π is computed by ∫π

−π

sin x dx. TRUE FALSE

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 5

(e) (4 points) To estimate the area below the function y = −x2 + 4 between x = 0 andx = 2, we may write a Riemann sum

Sn =

n∑i=1

(−

(2i

n

)2

+ 4

)· 2

n.

This sum Sn is less than the actual area for all n. TRUE FALSE

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 6

Question 3. (20 points) Consider the function

f(x) =1 − x2

x2 − 4.

We will get you started by computing the first two derivatives:

f ′(x) =6x

(x2 − 4)2and f ′′(x) = −

18x2 + 24

(x2 − 4)3.

(a) Find all critical and singular points of f.

(b) Find all asymptotes of f.

(c) Where is f increasing and decreasing?

(d) Where is f concave up and concave down?

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 7

(e) Using what you have calculated in the first four parts of this problem, please graphthe function

f(x) =1 − x2

x2 − 4

on the axes given below. There is a practice sheet of graph paper on the last sheet ofthis exam that you may wish to use to make a first attempt at the graph.

FINAL ANSWER:

x=1x=0

y=0

y=1

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Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 8

Question 4. (9 points) Suppose that a function f satisfies f(1) = 3 and its DERIVATIVE is

d

dx

(f(x)

)= e−x2

.

Please show your work when you answer the questions below.

(a) Find the linear approximation to f about x = 1.

(b) Use the linear approximation to estimate the value of f(1.5).

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 9

(c) Is you answer in (b) an under- or over-estimate? Please give an argument support-ing your answer.

(Hint: you do not need to compute the exact value of the error.)

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 10

Question 5. (15 points) A clown is filling a spherical balloon with helium from a tank thatemits helium at a rate of 250 cm3/s. Please answer the following questions, showing yourwork and specifying the units of your answers. You may wish to refer to the formulæ atthe end of this exam.

(a) How fast is the radius of the balloon changing when the radius is 5 cm?

(Hint: your answer should have a π in it.)

(b) How fast is the surface area of the balloon changing when the radius of the balloonis 5 cm?

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 11

(c) After 20 seconds, the balloon escapes the clown’s grip. Helium blows out of theballoon at a rate of 350 cm3/s. How fast is the radius changing when the radius ofthe balloon is 10 cm?

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 12

Question 6. (20 points) You are in the woods 4 km due north of the nearest point P on astraight east-west road. You need to get to the bus stop on the road that is 10 km west ofthe point P. You can walk 3 km/h in the wood and 5 km/h on the road. Please answerthe following questions.

(a) Draw a figure indicating your position, the point P, and the bus stop. Suppose thatyou walk to a point on the road, x km west of P. Write the travel time T (hours) as afunction of x (km).

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 13

(b) Suppose that the domain of T(x) is all real numbers. List the singular points andcritical points of T(x), and determine all local maximum and local minimum of T(x).

(Keep going!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 14

(c) To what point on the road between P and the bus stop should you head in order tominimize the travel time to the bus stop? Justify your answer.

(d) The last bus of the day leaves the stop in three hours. Can you make it, or will yoube sleeping under the stars? Please give evidence to support your answer.

(STOP – THIS IS THE END!!)

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 15

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 16

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Don’t forget to transfer your final work to the page where the question is posed!

Math 1110 (Spring 2009) Prelim 3 (04/21/2009) 17

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For practice in Problem 3:

x=1x=0

y=0

y=1

Don’t forget to transfer your final work to the page where the question is posed!