MATH 1040
Test 3
Spring 2015
Calc 2.6-2.8, 3.1-3.3
Version A
Student’s Printed Name: _______________________
CUID:___________________
Instructor: ______________________
Section # :_________
You are not permitted to use a calculator on any part of this test. You are not allowed to use any
textbook, notes, cell phone, laptop, PDA, or any technology on any part of this test. All devices
must be turned off while you are in the testing room.
During this test, any communication with any person (other than the instructor or his designated
proctor) in any form, including written, signed, verbal, or digital, is understood to be a violation
of academic integrity.
No part of this test may be removed from the testing room.
Read each question very carefully. In order to receive full credit, you must:
1.
Show legible and logical (relevant) justification which supports your final answer.
2.
Use complete and correct mathematical notation.
3.
Include proper units, if necessary.
4.
Give exact numerical values whenever possible.
You have 90 minutes to complete the entire test.
On my honor, I have neither given nor received inappropriate or unauthorized information
at any time before or during this test.
Student’s Signature: ________________________________________________
Do not write below this line.
Problem
Possible
Points
1
Points
Earned
Problem
Possible
Points
5
9a
6
2
4
9b
6
3
5
9c
6
4
9
10
6
5a
6
11
4
5b
6
12
6
6
6
13
4
7
4
14
11
8
6
Test Total
100
Points
Earned
Page 1 of 10
MATH 1040
Test 3
Spring 2015
Calc 2.6-2.8, 3.1-3.3
Version A
Read each question carefully. In order to receive full credit you must show legible and
logical (relevant) justification which supports your final answer. Give answers as exact
answers. You are NOT permitted to use a calculator on any portion of this test.
3
2
1. (5 pts.) For which x -value(s) does the function f (x) = 3x − 2x + 4 and g(x) = 2x + 0.5x
have the same slope.
Work on Problem
Points Awarded
f ′ ( x ) = 9x 2 − 2
Correct derivative
1 each
3
g ′ ( x ) = 6x 2 + x
Derivatives equal
0.5
Solves for x
2.5
Notes:
-1 algebra mistakes or not simplifying answer
-5 setting function = function or one function = derivative
-4 derivative = 0
only received 1 last point of the 2.5 points for solving if correct
9x 2 − 2 = 6x 2 + x
3x 2 − x − 2 = 0
( 3x + 2 )( x − 1) = 0
x=−
2
x =1
3
2. (4 pts.) Find the slope of the curve to y = 7sec x at x =
y′ = 7sec x tan x
π
.
4
Work on Problem
π
π
1
14
⎛π⎞
y′ ⎜ ⎟ = 7sec tan = 7 ⋅
⋅1 =
=7 2
⎝ 4⎠
4
4
2
2
2
Derivative
Evaluation
Correct answer
Notes:
-1 for y’(x)=y’(a)
-0.5 incorrect evaluation
Points
Awarded
2
1
1
3. (5 pts.) Sketch a possible graph of a function f (x) that satisfies all of the given conditions.
Be sure to identify all vertical and horizontal asymptotes with dotted lines.
f (1) = 0 ½ pt
lim f ( x ) = −1 ½ pt
x→ −∞
lim f ( x ) = 1.5 1 pt
x→1
lim f ( x ) = −1 ½ pt
x→ ∞
lim f ( x ) = ∞ ½ pt
x→2 −
lim f ( x ) = −∞ 1 pt
x→2 +
f is continuous everywhere except x = − 0.5, 1, 2
1 pt discontinuous at -0.5
answer is variable
-2 graph failed vertical line test, -1/2 each for not showing asymptote with dotted line (not
applied if limits were wrong)
Page 2 of 10
MATH 1040
Test 3
Version A
Spring 2015
Calc 2.6-2.8, 3.1-3.3
f ( x ) − f ( 3)
to find the slope of the tangent to the graph of
x→3
x−3
f ( x ) = x 2 + 3x + 1 at the point ( 3,19 ) .
4. a. (6 pts.) Use lim
x 2 + 3x + 1− 19
x 2 + 3x − 18
( x − 3)( x + 6 ) = lim x + 6 = 9
= lim
= lim
(
)
x→3
x→3
x→3
x→3
x−3
x−3
x−3
lim
Work on Problem
Points Awarded
Correct substitution
1
Correctly factors numerator
2
Correctly eliminates 0/0 issue
2
Correct limit (award only if correct)
1
Notes:
-2 pt for no limit notation or incorrect limit notation
-1 pt for poor limit notation
-1 two or more missing equals
-1/2 pt for limit notation carried too far
-2 “dropping” the denominator
-0.5 per algebraic error, -0.5 per term simplified incorrectly
-0.5 each skipping a step
• work that jumped from a correct point to the correct answer without showing work
in between - lose points for whatever steps were not shown
no penalty for not simplifying final fraction for slope
b. (3 pts.) Find an equation of the line tangent to the graph of f ( x ) = x 2 + 3x + 1 at the point
( 3,19 ) .
Work on Problem
Points Awarded
Correctly use the slope from part a
1
y − 19 = 9 ( x − 3)
Correctly uses the y-value
1
Correctly uses the x-value
1
Notes:
-1 if change to slope-intercept form incorrectly
-1.5 for solving y = mx + b incorrectly for b and never
writing a correct equation
-0.5 incorrect sign in the equation
-3 if equation is not a line, used f’(x)
no penalty for not simplifying final fraction for slope
Page 3 of 10
MATH 1040
Test 3
Spring 2015
Version A
Calc 2.6-2.8, 3.1-3.3
5. (6 pts. each) Evaluate the following limit. (Use of L’Hopital’s Rule is not permitted)
7x 3 − 9
lim
x→ −∞
−3x 3 + 64x 6 + 2
a.
1
1
9
7x 3 − 9 ⋅ 3
7− 3
3
7x 3 − 9
7−0
7
x = lim
x
x
lim
⋅
=
lim
=
=
−
x→ −∞
x→ −∞
−3 − 8
11
2
−3x 3 + 64x 6 + 2 1 x→ −∞ −3x 3 ⋅ 1 + 64x 6 + 2 ⋅ 1
−3 − 64 + 6
3
3
6
x
x
− x
x
(
)
Work on Problem
Points Awarded
Correct WORK for negative infinity
2
on terms that are not in the root
Correct WORK for negative infinity
2
on terms that are in the root
Limit as go to infinity applied
1
Correct answer (award only if correct)
1
Notes:
-1 including 0/0 in work
-1 missing equals (-0.5 for only one missing equals)
-1 missing limits (-0.5 for only one missing)
Excused 1 missing = sign ONLY if it was at a line break
-0.5 limit notation carried too far
-3 sqrt(a^2-b^2)=a-b
⎛ p2 − 4 ⎞
lim tan −1 ⎜ 2
p→ 2
⎝ 2 p − 4 p ⎟⎠
b.
⎛
⎛ p2 − 4 ⎞
( p − 2)( p + 2) ⎞ = tan −1 ⎛ lim p + 2 ⎞ = tan −1 ⎛ 2 ⎞ = tan −1 1 = π
−1
lim tan −1 ⎜ 2
=
tan
lim
⎜
⎟
⎜⎝ p→2 2 p ⎟⎠
⎜⎝ 2 ⎟⎠
p→ 2
4
⎝ 2 p − 4 p ⎟⎠
⎝ p→2 2 p ( p − 2 ) ⎠
Work on Problem
Points Awarded
Composition of limit
1
Factoring and cancelling
2
Substitution
1
Evaluation as p goes to a
1
Correct answer (award only if correct)
1
Notes:
-1 including 0/0 in work
-1 missing equals (-0.5 for only one missing =)
-1 missing limits (-0.5 only one missing)
-0.5 limit notation carried too far
Excused 1 missing = sign ONLY if it was at a line break
-1 for stating more than one solution to arctan(1)
-6 replacing arctan(a/b) with tan(b/a)
Page 4 of 10
MATH 1040
Test 3
Spring 2015
Version A
Calc 2.6-2.8, 3.1-3.3
6. (6 pts.) Find the x -value(s) of the point(s) where the graph of f (x) = cos x sin x has
horizontal tangent lines on the interval [0, 2π ] .
Work on Problem
f ′ ( x ) = cos x cos x − sin x sin x
Correct derivative
= cos 2 x − sin 2 x
function has horizontal tangents when derivative is 0
cos 2 x − sin 2 x = 0
sin 2 x = cos 2 x
sin x = cos x
π 5π
x= ,
4 4
or sin x = − cos x
3π 7π
x=
,
4
4
Points Awarded
2
1
3
Derivative = 0
Solves for x
Notes:
-0.5 to -1 algebra mistakes or not simplifying answer
-6 setting function = 0
only received last point of the 3 points for solving if correct
-0.5 each extra “solution”
-1 one incorrect final answer
-1.5 only presenting one solution
7. (4 pts.) List the values of 𝑥 at which 𝑓 is not differentiable. For each x -value where f is
not differentiable, state the reason why not.
x=-2 is not continuous (hole,
removable discontinuity)
x=1 is sharp point
x=2 is not continuous (jump
discontinuity)
x=3 is not continuous (vertical
asymptote, infinity
discontinuity)
½ pt per point and ½ pt per reason, -1/2 each extra pt. or extra reason
do not have to be specific on reason for discontinuities (“not continuous” is sufficient)
Page 5 of 10
MATH 1040
Test 3
Spring 2015
Calc 2.6-2.8, 3.1-3.3
Version A
8. (6 pts.) Let f ( x ) = 4 + x + 5 .
Use the limit definition of the derivative to show that f ′(x) =
lim
(
4+ x+h+5 − 4+ x+5
h→0
= lim
h→0
= lim
h→0
h
(
h
x + h + 5 − ( x + 5)
x+h+5 + x+5
) = lim
h→0
)
= lim
h→0
h
(
1
2 x+5
.
x+h+5 − x+5 x+h+5 + x+5
⋅
h
x+h+5 + x+5
h
x+h+5 + x+5
)
1
1
=
x+h+5 + x+5 2 x+5
Work on Problem
Points Awarded
Conjugate
1
Simplify numerator
2
Cancel
1
Evaluation as h goes to 0
1
Correct answer (award only if correct)
1
Notes:
-1 including 0/0 in work
-1 missing equals (-0.5 for only one missing equals)
-1 missing limits (-0.5 for only one missing)
-2 no limits
Excused 1 missing = sign ONLY if it was at a line break
-0.5 limit notation carried too far
up to -1 for notational errors (such as omitting the parentheses in
the denominator or poor limit notation)
-1 for stating the conjugate correctly but not multiplying the
denominator by the conjugate (-2 total if remainder of problem is
correct, additional -1 for incorrect solution)
-0.5 lack of limit rewrite after cancellation
-6 using derivative in place of f(x)
Page 6 of 10
MATH 1040
Spring 2015
Version A
Calc 2.6-2.8, 3.1-3.3
9. (6 pts. each) Find the derivative of the following functions. Use appropriate notation to
denote the derivative. Simplify by combining like terms, reducing fractions, and removing
negative exponents from answers.
a. f ( x ) = x π + π e + x x
Work on Problem
Points Awarded
f ′( x) = π x
π −1
Test 3
3 12
+ 0 + x = π x π −1 + 1.5 x
2
1⎛
t⎞
b. g(t) = csct − ⎜ 5t 6 − ⎟
⎝
t
π⎠
1⎛
t⎞
1
g(t) = csct − ⎜ 5t 6 − ⎟ = csct − 5t 5 −
⎝
⎠
t
π
π
Derivative of each term
2 each
Notes:
-2 incorrectly labeling derivative or not labeling derivative
-1 notational errors
Work on Problem
Points Awarded
Derivative of first term
2
Distribute, derivative of next two terms
4
(OK with product rule if correct)
Notes:
-2 incorrectly labeling derivative or not labeling derivative
credit can be given for product rule but must be simplified
g ′ ( t ) = − csct cot t − 25t 4 + 0
c. h(x) =
h′ ( x ) =
(
tan x − x
x5 + 3
)
You do NOT have to simplify this derivative.
⎛5 3
⎞
x 5 + 3 ( sec 2 x − 1) − ( tan x − x ) ⎜ x 2 + 0 ⎟
⎝2
⎠
(
)
x5 + 3
2
Work on Problem
Points Awarded
Keep bottom
0.5
Correct derivative of top
2
Correct derivative of bottom
2
Hold top
0.5
Denominator squared
1
Notes:
-2 incorrectly labeling derivative or not labeling derivative
-6 f’/g’
-3 top of quotient rule backwards
-3 denominator squared completely missing
-1 each for missing parentheses
simplified answers were not checked for correctness
Page 7 of 10
MATH 1040
Test 3
Version A
10. (6 pts.) For f ( x ) =
d2 f
1
x
,
determine
+
5e
2x 2
dx 2
Spring 2015
Calc 2.6-2.8, 3.1-3.3
x=1
1
1
f ( x ) = 2 + 5e x = x −2 + 5e x
2x
2
1
1
f ′ ( x ) = ( −2 ) x −3 + 5e x = − 3 + 5e x
2
x
3
f ′′ ( x ) = − ( −3) x −4 + 5e x = 4 + 5e x
x
f ′′ (1) = 3 + 5e
Work on Problem
Points Awarded
first derivative
2
(1 point per term)
second derivative
2
(1 point per term)
Evaluation
2
Notes:
-2 incorrectly labeling derivative or not labeling derivative
-1 missing plus sign for product rule
-0.5 slight multiplication error in simplification
-0.5 for copy error
-3 if first derivative correct, but then rewritten incorrectly so that
second derivative is wrong (i.e., work was not followed from
incorrect rewrite to second derivative; work was followed for the
evaluation and simplification)
-0.5-1 simplifying incorrectly (depending on severity)
-0.5 notation mistake (i.e., f ' ' = f ' ' (a), mixing x and theta, etc.)
-0.5 missing ( )
11. (4 pts.) Find the limit. Step by step work MUST be shown. You will not be given any credit
for using L’Hopital’s Rule.
Work on Problem
Points Awarded
sin 2x
lim
Proper
constants
separated
from
limit
1
x→0 10x
sin 2x 1
sin 2x 2
sin 2x 1
1
= lim
= lim
= ⋅1 =
x→0 10x
10 x→0 x
10 x→0 2x
5
5
lim
Multiplication by 1
1
Placing k/k where needed
1
Limit (only if correct)
1
Notes:
-4 sinkt=ksint
-4 (sinkt)/t = sink
-0.5 to -1 algebra mistakes or not simplifying answer
12. (6 pts.) If g ( x ) = x ⋅ f ( x ) , where f (3) = 4 and f ′ ( 3) = −2 , find an equation of the normal
line to the graph of g ( x ) at the point where x = 3 .
g( x) = x ⋅ f ( x)
g ( 3) = 3⋅ f ( 3) = 3⋅ 4 = 12
g ′ ( x ) = x ⋅ f ′ ( x ) + 1⋅ f ( x )
g ′ ( 3) = 3⋅ f ′ ( 3) + 1⋅ f ( 3) = 3⋅ ( −2 ) + 1⋅ 4 = −6 + 4 = −2
mnor = −
1 1
=
−2 2
so equation of normal is y − 12 =
1
( x − 3)
2
Work on Problem
Function evaluation g(3)
Derivative
Derivative evaluation
Slope of normal (can be implied
in equation)
Equation of line
Points Awarded
1
2
1
0.5
1.5
Notes:
-1 poor notation
-2 incorrectly labeling derivative or not labeling derivative
-1 incorrect sign on slope of normal line
-0.5 arithmetic error
-3 normal line is not linear
-1 g’(x)=g(3) notation
Page 8 of 10
MATH 1040
Test 3
Spring 2015
Version A
Calc 2.6-2.8, 3.1-3.3
13. (4 pts.) Given the graph of f ( x ) shown below, sketch the graph of the derivative of f ( x ) on
the second axes.
answers may vary somewhat (particularly y-intercept may vary)
Work on Problem
Correct roots on derivative
Correct shape
Correct increasing/decreasing
Notes:
Points Awarded
2
1
1
Page 9 of 10
MATH 1040
Test 3
Version A
Spring 2015
Calc 2.6-2.8, 3.1-3.3
x − 3x + 2 ( x − 2 ) ( x − 1) x − 1
=
= 2 x≠2
x 3 − 2x 2
x2 ( x − 2)
x
a. (3 pts.) Does f ( x ) have a vertical asymptote at x = 0 ? YES or NO
Use a limit to prove your answer.
14. Consider the function f ( x ) =
lim
x→0
x −1
−1
=
= −∞
2
x
small +
2
Work on Problem
Points Awarded
Shows evidence of understanding that x^2
2
gets small
Recognizes limit goes to positive infinity
1
Notes:
-0.5-1 poor notation (-0.5 if 0 is ever written in denominator; -1 if
no limit is written in problem; -1 if there is no limit of function as
first step)
-1 specific x-values plugged in
-2 if two one-sided limits were computed and one limit was
computed incorrectly
-2 if only one side of the two-sided limit is shown (even if it is
totally correct)
-3 incorrect sign on infinity
b. (3 pts.) Does f ( x ) have a vertical asymptote at x = 2 ? YES or NO
Use a limit to prove your answer.
Work on Problem
x −1 1
Uses factored form (or gets to the
lim 2 =
factored form)
x→2 x
4
Evaluation as x goes to a
Points Awarded
1
1
Correct limit (award only if correct)
1
Notes:
-0.5-1 poor notation (-1 if no limit is written in problem; -1
if there is no limit of function as first step)
-0.5 if final answer is correct but not simplified
c. (5 pts.) f ( x ) has a horizontal asymptote at both ends. Use a limit to find it. State the equation
of the horizontal asymptote. (Work must be shown – shortcuts cannot be used)
1
1 3 2
Work on Problem
Points Awarded
− 2 + 3 0−0+0
2
3
proper WORK for limit to infinity
3
x − 3x + 2 x
x
x
x
lim 3
⋅
=
lim
=
=
0
x→∞ x − 2x 2
1 x→∞
2
Finds the overall limit (award only if
2
1− 0
1−
correct)
3
x
x
Notes:
y = 0 is horizontal asymptote
-0.5 for not writing limit with all terms simplified after
correct division (notated as "show step")
-2 if not multiplying by a form of 1 (i.e., (1/x^3)/(-1/x^3))
-0.5 if one sign or term is incorrect
-0.5 for not simplifying 0/1
-0.5-1 poor notation
Page 10 of 10