Review for Test 2.7-3.2

Find an equation of the tangent line to the curve at the point (2, 1). 3( ) 5 3f x x x

1 7( 2)

7 13

y x

or

y x

The position function of a particle is given by

When does the particle reach a velocity of 166 m/s?

t=8seconds

3 2( ) 1.5 2 , 0s t t t t t

Determine the equation of the tangent line if

y+4=4(x-3) or y=4x-16

(3) 4, '(3) 4f and f

Find the points on the curve where the tangent is

horizontal.

(1,-6) (-2,21)

3 22 3 12 1y x x x

If a ball is thrown vertically upward with a velocity of 200 ft/s, then its height after t seconds is

What is the maximum height reached by the ball?

1000ft

2( ) 200 10s t t t

Find , if

'(0)f ( ) 1 4sinf x x

'(0) 4f

Sketch the graph of a continuous function with f(0)=1 and 2, 2

'( )1, 2

xf x

x

Using one-sided derivatives, show that the function does have a

derivative at x=1.

2 , 1( )

3 2, 1

x x xf x

x x

2' '

1 1

1 1

1 1

' '

2 3 2 1(1) lim (1) lim

1 1

( 1)( 2) 3( 1)lim lim

1 1lim 2 lim3

(1) 3 (1) 3

x x

x x

x x

x x xf f

x x

x x x

x xx

f f

Determine when the following particle is speeding up .3 2( ) 2 21 60 3f x t t t

72, (5, )2

and

Determine the displacement of the particle during the first two

seconds.

- 4 units It is four units to the left of where it started!!

2( ) 4 3s t t t

Sketch the graph of a function where

(2) 3, '(3) 4, '(4) 6, '( 1) 2f f f and f

If an arrow is shot upward on the moon with a velocity of 55m/s, it height in meters after t seconds is given by .

Find the average velocity over the interval[1,1.04].

54.9184

2( ) 55 .04h t t t

Find the derivative of .

2 3( ) ( 1)( 3)f x x x

4 2'( ) 5 3 6f x x x x

Write the equation of the normal line to the curve at x=0.

4 siny x x

1

5y x

Find the equation of the tangent line to the graph of at x=0. ( ) sinf x x x

0y

Determine the slope of the normal line to the graph at x=1.

1( )

1

xf x

x

'(1) 2f

Find for

2

2d y

dx

3

1

xy

x

38

1x