Example 1: Find the general antiderivatives of each of the following using you knowledge of how to find derivatives. a) 2 f x x b) f x x c) 4 7 2 3 F x x d) 2 1 g x x e) cos dy x dx Example 2: Find all functions g such that g x 4 2 4sin x x x x x . Example 3: Solve the differential equation f x 3x 2 if f 2 3 . Find both the general and particular solutions.
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4 cos 2 - Uplift Education · a) f x x 2 b) f x x c) 4 2 7 3 F x x d) 2 1 g x x e) cos dy x dx Example 2: Find all functions g such that g x 2 4 4sin x x x x x . Example 3: Solve
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Example 1:Find the general antiderivatives of each of the following using you knowledge of how to find derivatives.
a) 2f x x b) f x x c) 4
72
3F x x d) 2
1g x
x e) cos
dyx
dx
Example 2:
Find all functions g such that g x42
4sinx x
xx
x .
Example 3:
Solve the differential equation f x 3x2 if f 2 3 . Find both the general and particular solutions.
Example 4:
Find the particular solution to the following differential equation if 12x 20 1dy
e xdx
and y 0 2 .
Example 5:
Find the particular solution to the following differential equation if 2
22
12 6d y
xdx
x 4 and
a) y1 3 and y 0 6 b) y 0 4 and y 1 1.
Example 6:
a) Evaluate2
sin
cos
xdx
x b) Evaluate tan2 p 4dp
Show all work. No Calculator
Multiple Choice
1. If 2( ) 12 6 1f x x xʹ′ = − + , (1) 5f = , then (0)f equals(A) 2 (B) 3 (C) 4 (D) –1 (E) 0
2. Find all functions g such that ( )25 4 5x xg xx
+ +ʹ′ =
(A) ( ) 2 42 53
g x x x x C⎛ ⎞= + − +⎜ ⎟⎝ ⎠
(B) ( ) 2 42 53
g x x x x C⎛ ⎞= + + +⎜ ⎟⎝ ⎠
(C) ( ) ( )22 5 4 5g x x x x C= + − + (D) ( ) 2 4 53
g x x x x C⎛ ⎞= + + +⎜ ⎟⎝ ⎠
(E) ( ) ( )25 4 5g x x x x C= + + +
3. Determine ( )f t when ( ) ( )2 3 1f t tʹ′ʹ′ = + and ( )1 3f ʹ′ = , ( )1 5f = .
(A) ( ) 3 23 2 2 2f t t t t= − + + (B) ( ) 3 22 2 4f t t t t= − + +
(C) ( ) 3 23 2 3f t t t t= + − + (D) ( ) 3 2 2 3f t t t t= − + +
(E) ( ) 3 2 2 5f t t t t= + − +
4. Consider the following functions:
I. ( )2
1sin2xF x =
II. ( )2cos24xF x = −
III. ( )2
3cos2xF x = −
Which are antiderivatives of ( ) sin cos ?f x x x= (Hint: take the derivative of each and manipulate) (A) II only (B) I only (C) I & III only (D) I, II, & III (E) I & II only
5. A particle moves along the x-axis so that its acceleration at time t is ( ) 8 8a t t= − in units of feet andseconds. If the velocity of the particle at 0t = is 12 ft/sec, how many seconds will it take for theparticle to reach its furthest point to the right?