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Test 2 with Solutions for Calculus I | MATH 171, Exams of Calculus

Material Type: Exam; Professor: Jernigan; Class: Calculus I; Subject: Mathematics; University: Community College of Philadelphia; Term: Unknown 1989;

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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Math 171 Test 2 Solutions
Find the following derivatives. Simplify if possible, but don’t do anything silly.
2. sin
d
x
dx ⎡⎤
⎣⎦
Chain Rule: 1c
cos
2sin 2sin
os
x
x
x
x
×=
3.
()
cos 2
d
x
dx
Chain Rule again:
(
)
(
)
sin 2 2 2sin 2
x
x−×=
4. 2
x
de
dx
Another Chain Rule: 22
22
xx
exxe
×− =
5.
[
ln sin
d
]
x
x
dx Produce Rule: 1sin
sin ln cos ln cos
x
xx x
xx
×+× = + x
6. 2
2
dx
dx x
+
⎣⎦
Quotient Rule:
()
(
)
() () ()
22
21 21 22 4
22
xxxx
xx
−×−+× −−−
==
−−
2
2
x
7. sin x
dx
dx
Logarithmic Differentiation:
[]
sin sin sin
sinln lncos
xx
dx
x
xx x x x
dx x
⎛⎞
×=×+
⎜⎟
⎝⎠
Note that this was the answer to question #5.
8. sin 2
sin 2
dx
dx x
+
⎣⎦
Same as Question #4 with x replaced by sinx. Therefore, by the Chain
Rule and the answer to #4:
() ()
22
44
cos
sin 2 sin 2
cos
x
x
xx
−−
×=
−−
9. Find the first and second derivatives of the function () sin
f
xxx
=
.
Product Rule:
()
1sin cos sin cos
f
xxxxxx
+ = + x
Product Rule again:
()
cos 1 cos sin sin 2cos
f
xx xxxxx
′′ =+×+×= +x
pf3

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Math 171 Test 2 Solutions

Find the following derivatives. Simplify if possible, but don’t do anything silly.

  1. sin

d x dx

Chain Rule:

1 c cos 2 sin 2 sin

os x x x x

× =

3. cos 2( )

d x dx

Chain Rule again: − sin 2( x ) × 2 = −2sin 2( x )

d (^) x^2 e dx

⎡^ −

Another Chain Rule:

2 2 2 2

x x e x xe

− − × − = −

5. [ln sin

d

x x ]

dx

Produce Rule:

1 sin sin ln cos ln cos

x x x x x x x

× + × = + x

d x

dx x

Quotient Rule:

2 2

x x (^) x x

x x

− × − + × − − − −

2 x − 2

d (^) sin x x dx

Logarithmic Differentiation:

[ ]

sin sin sin sin ln ln cos

x d^ x x x x x x x x dx x

× = × ⎜ + ⎟

Note that this was the answer to question #5.

sin 2

sin 2

d x

dx x

⎡^ +

Same as Question #4 with x replaced by sin x. Therefore, by the Chain

Rule and the answer to #4:

2 2

cos sin 2 sin 2

cos x x x x

× =

  1. Find the first and second derivatives of the function f ( ) x = x sin x.

Product Rule: f ′ ( x ) = 1 × sin x + x cos x = sin x + x cos x

Product Rule again: f ′′ ( x ) = cos x + 1 × cos x + x × − sin x = − x sin x + 2 cos x

  1. Explain in clear English why the derivative of the function ( ) 10 is not

x f x =

1 10

x x

− ×

(i.e. why the power rule does not apply here.)

Because the variable is in the exponent!

  1. What is the derivative of ( ) 10?

x f x =

Logarithmic differentiation or memory:

( ) 10 [ ln10]^10 ln

x d x f x x dx

′ = × = ×

  1. What is the derivative of log 10 x?

x ln

  1. Find the linear approximation to the function

f x x

at a = 2

2 (^ )^ (^ )

f = = f ′ = − = − ⇒ L x = + − x − = − + = 3 − − (^) −

x x

  1. Find the slope of the line tangent to the curve x + y = 5 at the point (4,9).

y y y y x y y x y y x x

+ ′^ = ⇒ ′^ = − ⇒ ′^ = − = = ′= − = −

  1. Use the fact that

1 2

tan ( ) 1

d x dx (^) x

− ⎡ ⎤ = ⎣ ⎦

and the chain rule to find

1 tan ( )

d (^) x e dx

− ⎡ ⎤ ⎣ ⎦

2 2

x x x x

e e e e

× =

16. Let f ( x ) = sin x g , ( x ) = ln x h x , ( )= x

cos , , 2

sin ln

f x x g h x x (^) x

f g h x x

D D =

17. Use the chain rule to find the derivative of f D g D h x ( )

1 1 cos ln(^ )

cos ln 2 2

x x x x x

× × =