Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

Student’s Printed Name: ________________________________

CUID: _______________

Instructor: ____________________________________________

Section: ______________

Instructions: You are not permitted to use a calculator on any portion of this test. You are not allowed to use a textbook, notes, cell phone, computer, or any other technology on any portion of this test. All devices must be turned off and stored away while you are in the testing room. During this test, any kind of communication with any person other than the instructor or a designated proctor is understood to be a violation of academic integrity. No part of this test may be removed from the examination room. Read each question carefully. To receive full credit for the free response portion of the test, you must: 1. 2. 3. 4.

Show legible, logical, and relevant justification which supports your final answer. Use complete and correct mathematical notation. Include proper units, if necessary. Give answers as exact values whenever possible.

You have 90 minutes to complete the entire test.

Do not write below this line.

Free Response Problem

Possible Points

Free Response Problem

Possible Points

1.

10

5. a.

2

2.

7

5. b.

5

3. a.

5

5. c.

2

3. b.

5

6. (Scantron)

1

4. a.

3

Free Response

58

4. b.

5

Multiple Choice

42

4. c.

5

Test Total

4. d.

3

4. e.

5

Points Earned

Points Earned

100

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

Multiple Choice. There are 14 multiple choice questions. Each question is worth 3 points and has one correct answer. The multiple choice problems will count 42% of the total grade. Circle your choice on your test paper. 1. (3 pts.)

∫ f ( x) dx = 8 5

Assume

0

∫ f ( x) dx = −1 . 2

and

Find

0

∫ f ( x) dx = 7

5

2

a)

b)

5

5

∫ f ( x) dx = 9

∫ f ( x) dx = −7

5

2. (3 pts.)

∫ f ( x) dx = −9 2

2

c)

∫ f ( x) dx . 2

2

d)

5

Solve the initial value problem.

dv 1 = sec t tan t , v(0) = 1 dt 2 a) v(t ) = 2 cos t − 1 c) v(t ) =

1 1 cos t + 2 2

b) v(t ) = 2sec t − 1 d) v(t ) =

1 1 sec t + 2 2

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MATH 1060 Calculus of One Variable I

3. & 4.

3. (3 pts.)

Test 3 Version A KEY

The graph of h ( x ) is shown below. It consists of two straight lines. Use it to answer problems 3 and 4.

∫ 2h( x) dx 1

−2

∫ 2h( x) dx = −3 1

a)

c)

∫ 2h( x) dx = −6 1

b)

−2

−2

∫ 2h( x) dx = 10

∫ 2h( x) dx = 5

1

1

d)

−2

4. (3 pts.)

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

−2

∫ h( x) dx 0

5

∫ h( x) dx = −24 0

a)

b)

5

5

∫ h( x) dx = −16

∫ h( x) dx = −24

0

c)

∫ h( x) dx = 16 0

5

0

d)

5

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

5. (3 pts.)

Use three equal-width right rectangles to estimate the shaded area.

a) 18

6. (3 pts.)

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

b) 9

Find the value of the definite integral ∫

e7

e

a) −6

b) 7

c) 27

d) 6

c) − 7

d) 6

1 dx . x

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

7. (3 pts.)

Find the value c that satisfies the conclusion of the Mean Value Theorem for the given function and interval.

f ( x) = x − 1 ,

[1, 5]

a) c = 3 c) c =

8. (3 pts.)

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

b) c = 2

1 2

d) MVT does not apply

∫ x

Let F ( x ) =

4 − t 2 dt . Which one of the following statements must be true about the

0

graph of F at x = 1 ? a) F is increasing and concave down at x = 1. b) F is increasing and concave up at x = 1. c) F is decreasing and concave down at x = 1. d) F is decreasing and concave up at x = 1.

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

9. (3 pts.)

Find the value of the definite integral ∫ 1

1 dx . 1 + x2 0

π 6

a)

10. (3 pts.)

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

∫

b) 0

π 4

c)

d)

π 3

The figure shows the areas of three regions bounded by the graph of f and the x-axis. Find

3

−4

∫

3

a)

∫

3

c)

−4

−4

7 f ( x ) dx .

7 f ( x ) dx = 2

∫

3

b)

7 f ( x) dx = 12

∫

3

d)

−4

−4

7 f ( x) dx = 84 7 f ( x) dx = 14

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MATH 1060 Calculus of One Variable I

11. (3 pts.)

12. (3 pts.)

Test 3 Version A KEY

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

Let f ( x ) be twice differentiable on the interval (−∞, ∞) . Suppose f (5) = 1 , f ′(5) = 0 , and f ′′( x ) > 0 for all x. Which one of the following must be true? a)

f is concave down on ( − ∞, ∞)

b)

f has an absolute minimum of 1

c)

f has an absolute maximum of 1

d)

f ′ is decreasing on ( − ∞, ∞ )

Determine where the function f ( x ) is increasing if the first derivative is f ′( x) =

a)

( −1, ∞ )

b) ( −∞, −1)

c)

( −∞, −1) ∪ (0, ∞)

d) ( −∞, 0)

Version

e3 x ( x + 1) . x2 + 1

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

13. (3 pts.)

Evaluate the definite integral by interpreting it in terms of area.

∫ (2 + 4

)

16 − x 2 dx

0

14. (3 pts.)

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

a) 8π + 8

b) 4π + 2

c) 4π + 8

d) 8π + 2

Find the x-values at the absolute extrema of f ( x) = ln x − x on the interval [e −1 , e 2 ] . a) absolute maximum at x = 1, absolute minimum at x = e 2 b) no absolute maximum, absolute minimum at x = 1 c) absolute maximum at x = 1, absolute minimum at x = e −1 d) no absolute extrema

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MATH 1060 Calculus of One Variable I

Test 3 Version A KEY

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

Free Response. The Free Response questions will count 58% of the total grade. Read each question carefully. To receive full credit, you must show legible, logical, and relevant justification which supports your final answer. Give answers as exact values. 1. (10 pts.) A rectangle is to be inscribed in a semi-circle of radius 2 meters (see figure). Find the largest area that the rectangle can have. In your work you should: • State the area function A to be optimized in terms of a single variable x. • State the domain of the function. • Show all work needed to find and verify the value of x that maximizes the area.

A = 2 xy , where x 2 + y 2 = 4 ⇒ y = 4 − x 2 (y < 0)

A( x ) = 2 x 4 − x 2

Domain: x ∈ [ 0, 2 ] or ( 0, 2 )

−1/2 1  A′( x ) = [2] 4 − x 2 + 2 x  ( 4 − x 2 ) ( −2 x )  2 

=2 4−x −

2 x2

2

= {Find critical numbers.}

A′( x ) = 0 when 4 ( 2 − x 2 ) = 0

x2 = 2

8 − 4x2

4 − x2

=

4 − x2

=

2 (4 − x2 ) − 2 x2 4 − x2

4 (2 − x2 ) 4 − x2

A′ DNE when

4 − x2 = 0

x2 = 4

x = ± 2 (x = 2 only solution in domain) x = ±2 (Neither value is a critical number.)

{Use one of the following methods to verify maximum area at x = 2.} • •

•

For a closed interval, using Closed Interval Method: A(0) = 0, A( 2) = 4, A ( 2 ) =0,

so A has absolute maximum at x = 2; or use First Derivative Test or Second Derivative Test Using First Derivative Test: A '( x ) > 0 on (0, 2) and A '( x ) < 0 on ( 2, 2),

so A has absolute maximum at x = 2; or use Second Derivative Test

−1/2 1  [ −8 x ] 4 − x 2 − ( 8 − 4 x 2 )  ( 4 − x 2 ) ( −2 x )  4 x x 2 − 6 ) < 0 on (0, 2), 2 = ( A′′( x ) = 2 3 2 4−x 4 − x2

so A has absolute maximum at x = 2.

A

( 2) = 2

2⋅ 4−

( 2)

2

(

)

= 2 2 ⋅ 2 = 2(2) = 4 m 2

The maximum area of the rectangle is 4 m 2 .

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MATH 1060 Calculus of One Variable I

Test 3 Version A KEY

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

Work on Problem: Points Finds an area function in terms of a single variable (1 pt for recognizing area as 2xy) 3 points States the domain of the area function 1 point Finds the critical number 3 points Verifies the absolute maximum 2 points States the maximum area, i.e. answers the question 1 point Notes: • Subtract ½ point for notation errors such as missing derivative notation with a maximum of 1 point deduction for all notation errors • Subtract ½ point for not showing ± when taking square root • Subtract ½ point for not including units with final answer • Subtract 1 point for drawing a number line with no labels nor justification • Subtract ½ point for showing verification of maximum, but no conclusion about finding the maximum

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MATH 1060 Calculus of One Variable I

Test 3 Version A KEY

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

2. (7 pts.) Set up and evaluate a definite integral for the area of the region above the x-axis and below the curve y =

Area = ∫

1

x ( 6 − 5 x ) on the interval [0, 1] (see figure).

x (6 − 5 x ) dx

= ∫ x1/2 (6 − 5 x ) dx 0

1

= ∫ ( 6 x1/2 − 5 x 3/2 ) dx 0

1

0

 6 x 3/2 5 x 5/2  = −   3 / 2 5 / 2 0 1

=  4 x 3/2 − 2 x 5/2 

1

= 4(1) − 2(1) = 4−2 =2 3/2

0

5/2

−0

Work on Problem: Points Sets up definite integral to find the area 2 points Finds antiderivative of integrand 3 points Completes the evaluation of the definite integral 2 points Notes: • Subtract ½ point for notation errors such as missing integral sign, missing differential, removing the integration notation too soon, not removing the integration notation when necessary, with a maximum of 1 point deduction for all notation errors

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

3. (10 pts.) Evaluate the limits. Use of L’Hôpital’s Rule must be indicated each time it is used, either symbolically or in words. No credit will be awarded without supporting work. a) (5 pts.) lim+ (1 + 2 x )

3

x

x→0

lim+ (1 + 2 x )

x →0

= lim+ e x →0

=e =e =e

x

ln  (1+ 2 x ) 

3

x



3   x  lim+ ln (1+ 2 x )   x →0 

   lim+ ( 3/ x ) ln (1+ 2 x )  x →0  3ln (1+ 2 x )    lim+  x  x →0 

L

= e 0 0

3

3⋅2    + 1 2x   lim+  1  x →0   

IF

=e =e

 6   lim+ 1+ 2 x   x →0   6   1+ 2⋅0 

= e6 Work on Problem: Points Rewrites using e and natural log function 1 point Moves the limit into the exponent of e ½ point Uses log property to simplify ½ point Expresses result as one fraction ½ point Applies L’Hopital’s Rule correctly (1 pt. for numerator, 1 point for denominator) 2 points Uses direct substitution to find final answer ½ point Notes: • Subtract ½ point for failing to indicate use of L’Hopital’s Rule • Subtract ½ point for notation errors such as missing or incorrect limit notation, inappropriately using limit notation after direct substitution, with a maximum of 1 point deduction for all notation errors (excluding errors indicating use of L’Hopital’s Rule) • Subtract ½ point for the incorrect statement “anything = an indeterminate form” • Subtract ½ point for indicating the wrong indeterminate form • Award full credit for applying other correct techniques o Define y as a function and take natural log of both sides o Define y as the limit in the exponent, determine the value of the limit, then raise e to that result

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MATH 1060 Calculus of One Variable I

3.

Test 3 Version A KEY

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

2 x − 2sin x x→0 x3

b) (5 pts.) lim

2 x − 2 sin x x →0 x3 L 2 − 2 cos x = lim 0 IF x → 0 3x 2 0 L 2sin x = lim 0 IF x → 0 6x 0 L 2 cos x = lim 0 IF x → 0 6 0 2 cos 0 = 6 1 = 3

lim

Work on Problem: Points Recognizes indeterminate form (explicitly or implicitly) ½ point Applies L’Hopital’s Rule correctly three times (2 pts for the 1st application, 1 pt for each 4 point of the next two applications) Uses direct substitution to find final answer ½ point Notes: • Subtract ½ point for failing to indicate use of L’Hopital’s Rule • Subtract ½ point for notation errors such as missing or incorrect limit notation, inappropriately using limit notation after direct substitution, with a maximum of 1 point deduction for all notation errors (excluding errors indicating use of L’Hopital’s Rule) • Subtract ½ point for the incorrect statement “anything = an indeterminate form” • Subtract ½ point for indicating the wrong indeterminate form

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

4. (21 pts.) Let f ( x) =

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

x2 + 4 x

a) (3 pts.) Determine the equation of the oblique asymptote on the graph of f ( x) .

x x x + 0x + 4 2

x2 4

Work on Problem: Uses long division to divide numerator of f by denominator States the equation of the oblique asymptote Notes: •

Points 2 points 1 point

⇒y=x b) (5 pts.) Determine the intervals on which f ( x) is increasing or decreasing. Be sure to show the calculation of the first derivative. Put your final answers in the appropriate spaces below. Increasing:

( −∞, −2), (2, ∞ )

f ( x) = x +

4 x 4 x2 − 4 ′ f ( x) = 1 − 2 = x x2 f ′ = 0 when x = 2, −2 f ′ DNE when x = 0 f ′( x ) > 0 when x < −2 f ′( x ) < 0 when − 2 < x < 0 f ′( x ) < 0 when 0 < x < 2 f ′( x ) > 0 when x > 2

Decreasing:

( −2,0), (0, 2)

Work on Problem: Points Finds the first derivative 2 points Finds values of x where f ′ = 0 and f ′ DNE 1 point States the intervals on which f is increasing 1 point States the intervals on which f is decreasing 1 point Notes: • Subtract ½ point for stating decreasing intervals as (–2, 2) because zero is not in the domain of f

c) (5 pts.) Determine the intervals on which f ( x) is concave up or down. Be sure to show the calculation of the second derivative. Put your final answers in the appropriate spaces below. Concave Up:

f ′′( x ) =

(0, ∞ )

8 x3 f ′′ is never 0 f ′′ DNE when x = 0 f ′′( x ) < 0 when x < 0 f ′′( x ) > 0 when x > 0

Concave Down:

( −∞, 0)

Work on Problem: Points Finds the second derivative 2 points Finds value of x where f ″ DNE 1 point States the interval on which f is concave up 1 point States the interval on which f is concave down 1 point Notes: • Subtract ½ point for incorrectly stating x = 0 is the solution of f ″(x) = 0

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

d) (3 pts.) Determine the equation of any vertical asymptotes on the graph of f ( x) . Support your answer with appropriate limits

lim f ( x )

x → 0−

= lim− x →0

x2 + 4 x

= −∞ ⇒x=0

lim f ( x )

x → 0+

= lim+ x →0

=∞

x2 + 4 x

Work on Problem: Finds at least one of the limits as x → 0– or x → 0+ States the equation of the vertical asymptote Notes: •

Points 2 points 1 point

e) (5 pts.) Sketch f ( x) . Show the ordered pair (x, y) at any point where f has a local extreme or an inflection point. Label all axis intercepts. Show the equation of any horizontal, vertical or oblique asymptotes on the graph.

Work on Problem: Oblique asymptote Vertical asymptote (Okay if x = 0 is not labeled) Local extrema Basic shape (increase/decrease, concavity, no intercepts) Notes: •

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Points 1 point 0 points 2 points 2 points

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

5. (9 pts.) Consider the limit below.

lim ∑ (6 − 2 xi* )∆x, where n

n →∞

i =1

•

the interval [0, 2] is partitioned into n subintervals of width ∆x =

•

xi* is the right endpoint of the ith subinterval in the partition

2 n

a) (2 pts.) Express the limit as a definite integral.

lim ∑ (6 − 2 xi* ) ∆x = ∫ ( 6 − 2 x ) dx n →∞

n

2

i =1

0

Work on Problem: Integrand Limits of integration (½ pt each) Notes: •

Points 1 point 1 point

b) (5 pts.) Using the summation formulas below as needed, evaluate the limit. c = cn, ∑ i =1

i= ∑ i =1

n

n

lim ∑ (6 − 2 xi* ) ∆x, xi* = n

n →∞

i =1

  2i   2 = lim ∑  6 − 2    n →∞  n  n i =1  n 2  4i  = lim ∑  6 −  n →∞ n n i =1 

2i 2 , ∆x = n n

n

2 4  = lim  ∑ 6 − ∑ i  n →∞ n n i =1   i =1 2 4 n ( n + 1)  = lim  6n − ⋅ n →∞ n n 2   2 = lim [ 6n − 2n − 2 ] n →∞ n 2 = lim [ 4n − 2 ] n →∞ n 4  = lim  8 −  n →∞ n  =8−0 =8 n

n

n(n + 1) , 2

i2 = ∑ i =1 n

i3 = ∑ i =1

n(n + 1)(2n + 1) , 6

n

n 2 (n + 1)2 4

Work on Problem:

Points 1 point

* i

Determines

x

Substitutes

xi* into sum

Substitutes ∆x into sum Uses summation formulas to get an expression in terms of n only Evaluates the limit (Okay if no work shown to ∞ resolve IF) ∞ Notes: •

Version

½ point ½ point 2 points 1 point

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Test 3 Version A KEY

MATH 1060 Calculus of One Variable I

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

c) (2 pts.) Evaluate the definite integral by using basic area formulas. Include a sketch.

∫ ( 6 − 2 x ) dx 2

0

= Area of rectangle + Area of triangle 1 = (2)(2) + (2)(4) 2 =4+4 =8

OR

∫ ( 6 − 2 x ) dx 2

0

= Area of trapezoid 1 (2)(6 + 2) 2 =8 =

Work on Problem: Sketches region Finds area with supporting work: area of rectangle + area of triangle OR area of trapezoid Notes: •

Version

Points 1 point 1 point

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MATH 1060 Calculus of One Variable I

Test 3 Version A KEY

Spring 2017 Sections 4.1 – 4.9, 5.1 – 5.3

Scantron (1 pt.) Check to make sure your Scantron form meets the following criteria. If any of the items are NOT satisfied when your Scantron is handed in and/or when your Scantron is processed one point will be subtracted from your test total. My Scantron:

□ is bubbled with firm marks so that the form can be machine read; □ is not damaged and has no stray marks (the form can be machine read); □ has 14 bubbled in answers; □ has MATH 1060 and my section number written at the top; □ has my instructor’s last name written at the top; □ has Test No. 3 written at the top; □ has the correct test version written at the top and bubbled in below my XID; □ shows my correct XID both written and bubbled in; Bubble a zero for the leading C in your XID.

Please read and sign the honor pledge below.

On my honor, I have neither given nor received inappropriate or unauthorized information at any time before or during this test. Student’s Signature: _____________________________________________________

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