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Test Chapter 3 Graphing M121 Form A

Name _______________________________ Date ________________

Show all work for credit.

1. Use the first derivative to find all relative extrema (max/min) for f(x) = 2x(x –4)3 . {you will

need to factor out the least powers) Show number sign chart Circle the (x,y) coordinates of

any answers

2. Find the points of inflection for y = 5×4 – x5

Show number line sign chart Circle answer(s)

3. Sketch and label (extrema and points of inflection) for the graph of a function having the

following characteristics:

f(0) = 2 f(2) = 0

f’(x) chart:

f(1) = 1 x = 3 is a vertical asymptote Horizontal asymptote: y=0

+

–

+

0

f’’(x) chart:

+

2

–

0

+

1

3

+

2

+

3

4. Sketch the Graph ( labeling all extrema and inflection points) (Show number line analysis

for f’(x) and f’’(x)}

f(x) = x4 – 4×3

Step 1 f’(x) =

Step 2 f’’(x) =

Step 3 Table

Step 4 sketch

!

5. Sketch: ( ) = ‘( + 1)! + 2

{label any extrema and points

of inflection}

= ___________ (rewrite)

Step 1 f ‘ =

f ‘ number line

Step 2 f ‘ ‘ =

f ‘ ‘ number line

Step 3 Table

Step 4 Sketch and Label

6. Find the Critical numbers of the first Derivative for =

(#$%)”

#”

7. Find the critical numbers for the second derivative when

first derivative is

′ = (

$!#

#$%)!

8. Find the absolute max/min of y = 2×3 – 9×2 +12x on the

interval [-1,4]

!”#

9. Sketch and label extrema and points of inflection: ( ) = ! !

Need number line analysis for both derivatives.

Vertical asymptote

x = ____

Horizontal asymptote y = _____

Step 1:

f’(x) =

$%(#$!)

f ‘ number line

f ‘ ‘ ( x)

=

!(#$’)

##

f ‘ ‘ number line

Step 3 Table

Step 4 Sketch and label

#!

show limit needed

10. Sketch and Label

: ( ) =

!# ”

(#$%)”

{Label extrema and points of inflection}

Note: The second derivatives is already given. You need to show work for the first

derivative. Use number line analysis for each derivative.

VA ________

HA _________ Show limit

Step 1:

Find ( = (

$)#

#$%)!

!

Step 2: Second derivative

(( ( ) =

)(!#*%)

(#$%)#

!!

Step 3: Table

Step 4 Sketch and label

11. Use implicit differentiation to find dy/dx for

when x = 1 and y = 2

{Uses product rule)

xy2 + y3 = 10

Bonus For the profit function ” − 10 # = ” , (p = profit in dollars) Find how rapidly the

profit P is growing when sales ( x = # items sold) is growing by 10 items per day and when the

number of items sold is 2 and when profit p = $4. Hint: take the derivative of both sides with

respect to t = time in days.

(write the units of the dp/dt in your answer)

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