(speed)Limits
Differentiate This
Two Chains Too Tangent
Xtreme Values and Local News
Straight Outta #2016Exams
100

Evaluate the limit as x approaches 2 of (x2/(x+3))

-4/5
100

2x3-6x-2

6x2-6

100

Calculate the average rate of change of f(x)= (x2)/2 - (1/x) on [2, 4].

25/8
100
Sketch an example of a function on a closed interval that has 2 local maximums,1 local min, an absolute max that occurs at a local extremum, and an absolute min that does not occur at a local extremum.
Answers Vary.
100
Find the equation of the tangent line to the graph of y=(x+1)/(2x) at x=1.
y-1= -.5(x-1)
200

Evaluate the limit as x approaches 3 of (x2-9)/(2x-6)

3
200

(x+1)(x2-1)

(1)(x2-1)+(x+1)(2x)

200

Find the equation of the tangent line to the graph of y=5x4-16x at x=1.

y+11=4(x-1)
200

Find and classify all local extrema of f(x)= 2x3-6x-3

Local Min at x=1, Local Max at x=-1
200

Find the location of all local extrema of f(x)= 3x4-12x3+12x2+17

There are local mins at x=0 and x=2, local max at x=1.
300

Evaluate the limit as x approaches 0 of x/(2x2 - x)

-1
300
sqrt(x)+1/sqrt(x)

0.5x-1/2-0.5x-3/2

300

Find y' for y=(x3-x)17

y'=17(x3-x)16(3x2-1)

300
Find the Absolute Extrema of f(x)=4x+(36/x) on the interval [1, 6].
Absolute Max at (1, 40) and absolute min at (3, 24)
300
Suppose that D(x) represents the depth of a certain lake (in feet) at a distance x feet from the shore. Explain what D(20)=5 and D'(20)=0.3 mean in practical terms for a person standing in the lake.
When you are 20 feet from the shore the depth is 5 feet. When you are 20 feet from the shore the lake is getting .3 feet deeper for each foot you move away from the shore.
400

Use the limit definition of the derivative (provided in class) to find f'(x) for f(x)= 2x-x2

2-2x
400

(4/3)x2+(1/6)x3.2-(2/(3x2))+4

(4/3)(2x)+(1/6)(3.2x2.2)-(2/3)(-2x-3)

400

Find y' for y= (x2-1)e(x2 - 1) 

y'=(2x)e(x2 - 1) +(x2-1)e(x2 - 1)(2x)

400

Find and classify all absolute extrema of f(x)= -2x3-x2+4x-1 on [-3, 1].

Absolute max at (-3, 32) and absolute min at (-1,-4)
400

Find f'(x) for f(x)= (1-x2)/(x4-x2-2)

f'(x)= [(x4-x2-2)(-2x)-(1-x2)(4x3-2x)]/(x4-x2-2)2

500
Use the limit definition of the derivative (provided in class) to find f'(x) for f(x)= x+(1/x)

1-(1/x2)

500

(ex)/(ln(x))

[ln(x)ex - (ex)/x]/[ln(x)]2

500
Evaluate (d/dx)[ln(ln(ln(x)))]
(1/ln(ln(x))) * (1/ln(x))* (1/x)
500

A company makes plastic buckets. One-hundred buckets sell for $75. The total cost in dollars of making x-hundred buckets is C(x)= 5x2 -20x+12. Assume that the company can sell all of the buckets it makes. How many buckets should be made to maximize profit.

x=9.5 HUNDRED, or 950 buckets.
500

Find the location of all local extrema of f(x)= 3x4/3-12x1/3. You must identify all critical points and use a derivative test to determine if it is a local max or min. 

There are CP's at x=0, x=1 and a min at x=1.
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