MAT 145: Test #3 (100 points)
Name ________________________ Calculator Used ____________
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Score
____________________
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Use the function f (x) = 4x − x + 2 for questions 6 through 9. Express all solutions using exact values.
6. Determine an equation for the tangent line to f when x = –1. Show all appropriate steps to justify your
result and express your equation in the form y = mx + b.
_______________________
7. Leo was asked about the domain and range of the function f. Leo claimed that the domain included all
real numbers. Explain how you know Leo is correct.
8. Ambrosia created the following limit statements about the function.
(a) Circle each TRUE limit statement.
(A) lim f (x) = 0
x→2
(B) lim f (x) = ∞
x→∞
(C) lim f (x) = 5
x→1−
(D)
lim f (x) = ∞
x→0
(b) Select ONE false limit statement from above and explain how you know it is a false statement. It is
not sufficient to simply state, “The limit is not correct.” Be sure to indicate which limit statement you are
discussing.
9. Zhiangi claims there are no points on the graph of f where the function reaches a maximum or a
minimum. Use your calculus knowledge of functions and their derivatives to support or refute Zhiangi’s
claim. Be clear, precise, and specific, referring to calculus-based evidence.
A particle moves along the x axis of a coordinate plane so that its position in relation to the origin, in
centimeters, at time t is s(t) =
1 4 3
t + t − 2t , t in seconds, t any real number. Use this information for
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questions 10 through 14. Please carefully check the units of measure you use with your responses!
10. Calculate the velocity function, v(t), at time t seconds. Include units.
_______________________
11. Calculate the acceleration function, a(t), at time t seconds. Include units. ______________________
12. Determine the instantaneous rate of change of the particle’s position at time t = 2 seconds. Include units.
_______________________
13. Calvin stated that the particle was moving to the left at precisely the time t = –1. Is Calvin correct?
Produce calculus-based evidence to support or refute Calvin’s statement and explain how you know based on
that evidence.
14. Rhonda and Fritz were discussing how to calculate the average velocity of the particle on the time
interval 0 ≤ t ≤ 2.
Fritz: The average velocity for this time period is just the average of v(0) and v(2):
v(0) + v(2)
.
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Rhonda: No way! The average velocity for this time period is the change in the particle’s position over
the change in time:
s(2) − s(0)
.
2
Is either student correct? Explain.
15. The function C(p) describes the total daily cost, in dollars, of operating a hospital when there are
p patients in the hospital, p ≥ 0. We know C(0) = $45,500 and C(24) = $178,700. Explain the meaning of
C!(10) = 3, 420 for this situation. In your explanation, be sure to indicate the appropriate units for C!( p) .
BONUS!
BONUS!
BONUS!
Carry out logarithmic differentiation to determine the derivative of the function
y = (sin(2x))
dy
x 2 cos(x)
.
Show all steps leading to expression of /dx in terms of x and simplify where appropriate.
Calculus I
MAT 145
Test #3: 100 points
Evaluation Criteria
Part I: No Calculators (30 points)
(1) – (4) 6 pts each; note the instructions provided with some derivative requests
(5) 6 pts (correct spelling, complete first and last name for each)
Part II: Calculators May Be Used (70 points)
(6) 6 pts (correct slope, correct equation, evidence included)
(7) 6 pts (complete, accurate and clear explanation for Leo’s claim)
(8) 8 pts (a) 4 pts (correct assessment of T/F of each limit statement; (b) 4 pts (correct, clear, and complete
explanation for a false limit statement)
(9) 10 pts (clear, precise, and specific response with complete and accurate calculus-based justification)
(10) 6 pts (correct response with appropriate units)
(11) 6 pts (correct response with appropriate units)
(12) 6 pts (correct response with appropriate units)
(13) 8 pts (correctly agree or disagree with Calvin; correctly use calculus-based evidence to support decision)
(14) 8 pts (clear and appropriate explanation with reference to claims made by Fritz and Rhonda)
(15) 6 pts (correct explanation, including justification based on calculus; correct units)