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2/8/09 Major Quiz 2 Review Problems Answers
1.) Find dx
dy for each of the following.
(a) y = 3x^3 − 5x^2 + 4x +13 (b) y = −3(x^5 + 2x^4 ) (c) (^) 2
(^5) x
x y = +
2
dy
x x
dx
4 3
dy
x x
dx
2
3
dy x
dx x
(d) (^) 4 5
x x
y = − (e)^2
1 2
1
2
− y = x − x (f) 1
x
x x y
5 6
dy 12 10
dx x x
dy
dx x x
2 2
dy x x x x
dx x
(g) y = (4x^2 − 1)(2x^3 + x + 5) (h) 1
3
3
−
x
x y (i) y = (2x^4 − 3x)(5x^2 − 2x + 5)
dy (8 )(2 x x 3 x 5) (4 x 2 1)(6 x 2 1)
dx
2 3 3 2
3 2
(3 )( 1) ( 1)(3 )
( 1)
dy x x x x
dx x
− − +
−
dy (8 x 3 3)(5 x 2 2 x 5) (2 x 4 3 )(10 x x 2)
dx
(j) (^36)
2
x
x y (k) y = 2 5
x
x (l) y = (^2)
3 4
5 4
x
x x
x − + − 3 2 2
3 2
(2 )( 6) ( )(3 )
( 6)
dy x x x x
dx x
2
(3)(2 5) (2)(3 1)
(2 5)
dy x x
dx x
4 2 5 3
dy
x x
dx x x
2.) Use the definition of derivative to find f'(x) for f(x) = 3x^2 + 5x − 1.
0
lim h
f x h f x → h
0
lim h
x h x h x x → h
2 2 2 0
lim h
x xh h x h x x → h
3.) Prove the following theorem.
"If f and g are functions and if k is the function defined by k ( x ) = f ( x ) + g ( x )
then if f '( x )and g '( x )exist, k '( x ) = f '( x ) + g '( x )."
0 0
( ) ( ) ( ) ( ) [ ( ) ( )]
'( ) lim lim h h
k x h k x f x h g x h f x g x k x → (^) h → h
0
lim h
f x h f x g x h g x → h
0
lim h
f x h f x g x h g x → h h
+^ −^ +^ −
0 0
lim lim h h
f x h f x g x h g x → (^) h → h
+^ −^ +^ −
= + = f '( ) x + g '( ) x
2 0
lim h
xh h h → h
0
lim h
h x h → h
6 x + 5
2/8/09 Major Quiz 2 Review Problems Answers
4.) Find an equation of each of the lines through the point (1, 2) that is tangent to the curve y = 2x^2 + 8.
y ' = 4 x y '(1) = 4(1) = 4 y − 2 = 4( x −1) y = 4 x − 2
5.) Find the values of a and b such that f is differentiable at 2 if f(x) = (^)
x^2 if x
ax b if x .
The derivative of f when x<2 is a, and 4x when x > 2. For the derivative to exist at 2, the pieces must be equal when x = 2. So, combining the two pieces for x = 2, a = 4(2). So, a = 8. Then, to find b, recall that the function must be continuous if it is going to be differentiable. So, the two pieces must be equal when x = 2. Using that information along with a = 8 and x = 2, then... 8(2) + b = 2(2)^2 - 1. So, b = -9.
6.) Find an equation of the tangent line and normal line to the curve y = 3x^4 − 12x at the point (1, −9).
y ' = 12 x^3 − 12 y '(1) = 12(1)^3 − 12 = 0 y − −( 9) = 0( x −1) y = − 9
7.) Use the definition of derivative to find f'(x) for f(x) = 2x^2 − 3x + 1.
0
lim h
f x h f x → h
0
lim h
x h x h x x → h
2 2 2 0
lim h
x xh h x h x x → h
8.) Find f '( x )for each of the following.
(a) f(x) = 5x^3 + 2x^2 − 3x (b) f(x) = (3x + 5)(2x^2 − 3x +1) (c) f(x) = 3 7
x
x
f '( ) x = 15 x^2 + 4 x − 3 2
f '( ) x = (3)(2 x − 3 x + 1) + (3 x + 5)(4 x −3)
4 5 2
x x x
f x
x
(d) f(x) = (^42)
x x
− (e) f(x) = 2 4
x
x
5 3
f '( ) x
x x
2 2
x x x
f x
x
9.) Use the definition of derivative to find f 'for f ( x ) = 3 x^2 − 5.
0
lim h
f x h f x → h
0
lim h
x h x → h
2 0
lim h
xh h h → h
0
lim h
h x h → h
4 x − 3
2 0
lim h
xh h → h
0
lim h
h x h → h
6 x
2 2 2 0
lim h
x xh h x → h
2/8/09 Major Quiz 2 Review Problems Answers
17.) Using the position function
s t ( ) 4 t – t
= , find the velocity function.
1 s '( ) t = 2 t^ −^2 + 3 t −^2
18.) Below is a graph of f ( ) x. Sketch a graph of f ′( ) x.
19.) Compute the derivative f ‘ ( x ) of
f x x
2 2
x f x x x
20.) Find an equation of the line tangent to f ( ) x = 7 x – 5 x – 5at x = 5. 1 '( ) 7 2 – 5 '(5) 7 – 5 (^2 2 )
f x x f
− = = If x = 5, f (5) = 7 5 – 30.
y x
21.) Find the derivative of ( ) 2
x + 2 x –5 x + x
2 2
x x x x x
x x
22.) Below is a graph of f ′( ) x. Sketch a graph of a plausible f ( ) x.
23.) Find the derivative of (^2)
x f x x
2 2 2
x x x f x x
2/8/09 Major Quiz 2 Review Problems Answers
24.) Find an equation of the line tangent to h x ( ) = f ( ) x g x ( ) at x = 3 if
f (3) = –3 , f ′(3) = 3 , g (3) = 3 , and g ′(3) = –1. h '( ) x = f '( ) x g x ( ) + f ( ) x g '( ) x h (3) = f (3) g (3) = −( 3)(3) = − 9 h '(3) = f '(3) g (3) + f (3) g '(3) h '(3) = (3)(3) + −( 3)( 1) − = 12 y − − ( 9) = 12( x −3)
25.) Find an equation of the line tangent to
f x h x g x
= at x = 1 if
f (1) = –3 , f ′(1)^ = –1, g (1) = 1 , and g ′(1)^ = 2.
2
[ ( )]
f x g x f x g x h x g x
f h g
2 2
[ (1)] (1)
f g f g h g
y − − ( 3) = 5( x −1)
26.) Below is a graph of f ′( ) x. Sketch a graph of a plausible f ( ) x.
Answers may vary. Below is one possible answer.
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