MAT 145
Semester Exam Part 1
Name ______________
100 points (Part I: 50 points)
Impact on Course Grade: approximately 25%
Do Not Use a Calculator!
Score ______________
Part I: Do Not Use Any Calculator or Computer Tools!
(1) – (10): State the derivative of each function. Each should start with y ! = or
dy
= . (2 pts each)
dx
y = 4x 5
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y = xe12
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y = 7x 6 − 5x 7
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y = 7x
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y = sec x
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( )
y = 3sin 2x
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y = ln ( x )
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y = x 3e−4
___________________________________
y=
8x 8 − 6x 6
2x 6
y = 2e x
___________________________________
___________________________________
Part 1 Continued: Do not use calculators or any other technology tools!
N
Name
____________________________________________________________________________
Questions (11) through (15) are each worth 2 points.
3
11. Calculate the derivative of h(x) = 3e 5x .
12. Calculate the derivative of
e x −1
. Simplify.
1+ e x
_______________
_______________
For questions 13 through 15, determine the exact value of each integral. Show
appropriate evidence to support your responses.
3
13.
∫ x 2 dx
__________________________
0
€
8
14.
∫(
1
15.
€
"
#
3
)
x 2 dx
1%
x&
∫ $ 3e x + ' dx
__________________________
__________________________
Part 1 Continued: Do not use calculators or any other technology tools!
( )
2
16-(i) Consider the function f x = 3x sin x . Which of the following responses is the
()
most accurate? (2 pts)
(A)
(C)
(E)
()
( )
f ! ( x ) = 3sin ( x ) + 6x cos ( x )
f ! ( x ) = 3sin ( x ) + 6x cos ( x )
f ! x = 3cos x 2
2
2
2
2
2
(B)
(D)
()
( )
( )
f ! ( x ) = 3x cos ( x ) + 6x sin ( x )
f ! x = 3x cos x 2 + 3x 2 sin x 2
2
2
(F) None of (A) through (E) is correct.
2
16-(ii) Apply implicit differentiation to the relation 2xy − 3y = 4x y and use that result
to determine the slope of the non-horizontal tangent line at x = 1. (3 pts)
2
17. A particle moves along the x-axis with velocity v at time t, t in minutes, for 0 ≤ t ≤ 6,
t2
given by v(t) = − 2t + 2 . At time t = 0, the particle is at x = 1. Show evidence for all
2
responses. Check labels! (1 pt each)
(a) Calculate the velocity of the particle at time at t = 4. _______________________
(b) At time t = 1, what is the particle’s acceleration?
_______________________
(c) Determine the time, for 0 ≤ t ≤ 6, when the particle first changes direction.
_______________________
(d) On the interval 0 ≤ t ≤ 6, at what time, if any, does the maximum velocity occur?
_______________________
(e) Use the information here to determine the position function, s(t).
_______________________
Part 1 Continued: Do not use calculators or any other technology tools!
f(x) = –
18-(i) Refer to the function g(x) plotted below, on the interval −4 ≤ x ≤ 10 , to determine
the following values. The function consists of three line segments and a semi-circle. Your
9 – (responses
x – 3) + 3
should be exact values. (4 pts)
2
8
h(x) = 1.5·x – 15
q(x) = – 3·x + 21
r(x) = 0.74·x + 3
7
0
∫
10
g(x) dx =
∫ g(x) dx =
6
6
−4
5
7
∫
g(x) dx =
8
6
∫ g(x) dx =
4
0
3
2
1
–8
–6
–4
y = g(x)
–2
2
4
6
8
10
–1
–2
–3
x
18-(ii) Let G(x) =
∫ g(t) dt
for −4 ≤ x ≤ 0 , with the graph of g(x) shown above.
–4
−4
Determine a symbolic representation for G(x) assuming that G(–4) = 0. (1 pt)
_______________________
19. Use the definition of derivative to determine f "( x ) for f (x) = x +1 . To earn ANY
credit for your solution, you must show appropriate algebraic evidence leading from the
definition of derivative for this function to a final simplified derivative function. (5 pts)
€
_______________________
Part 1 Continued: Do not use calculators or any other technology tools!
8
()
2
f(x) =
·x + 0.8
5
π
g(x) = 2.25·sin
·x – 0.21 – 0.06
7
h(x) = 2
q(y) = –3
r(y) = 3
s(y) = 4
(( )
7
)
6
BONUS #1
5
4
3
(3,2)
2
y = f(x)
1
P
12
– 10
–8
–6
–4
Q
–2
2
4
6
8
10
–1
–2
–3
–4
The figure shows the graph of a differentiable function y = f (x) and the tangent line to
that graph at the point (3,2). The function f(x) has horizontal tangents at x = –3 and x = 4.
–5
–6
x
The function g(x) is defined as g(x) =
∫
–7
f (t) dt . Suppose we know that g(3) =
0
11
.
5
Provide complete and appropriate evidence for your responses.
Calculate g !(3) and g !!(3) . (2 pts)
For what values of x on the open interval –7 < x < 7 does g have a relative
maximum? Justify. (3 pts)
(iii) For what values of x on the open interval –7 < x < 7 does g have a point of
inflection? Justify. (3 pts)
(i)
(ii)
BONUS #2
The function h(x) = 2xe
2x
is defined for all real numbers x.
(i) Calculate the location and exact value of the absolute minimum of h(x). (2 pts)
(ii) Suppose y = bxe bx , with b a non-zero constant. Show that the absolute minimum
value of y remains the same for any non-zero constant b. (3 pts)