Quick Review

Simplified Difference Quotient

100

Find the simplified form of the difference quotient for f(x)=-2x²-3x.

-2hx-2h-3

200

For  F(x)=-3x²+2,  find a simplified form of the difference quotient. Next, use the simplified form to find the difference quotient value when x=-1 and h=-1.

 DQ=9

300

For  F(x)=x²-6x+2, find a simplified form of the difference quotient. Next, use the simplified form to find the difference quotient value when x=1.5 and h=0.05.

DQ=-2.95

400

For  F(x)=-3/x  find a simplified form of the difference quotient. Next, use the simplified form to find the difference quotient value when x=3.5 and h=-2.5.

DQ=0.86

Utilizing the definition of derivative

100

What is the definition of a derivative?

f'(x)=lim f(x+h)-f(x)/h

       h->0

(couldn't add a picture to make the notation better, please excuse me)

200

For the function F(x)=x²+2x-5, find f'(x) by definition.

f'(x)=2x+2

300

For the function F(x)=x²-2x+3, find f'(x) by definition. Next, compute f'(-2).

f'(x)= 2x-2

f'(-2)= -6

400

For the function F(x)=-1/4x², find f'(x) by definition. Next, compute f'(-1).

f'(x)=-1/2x

f'(-1)=1/2

Equation of a Tangent line

100

Define a tangent line. Give a visual example of a tangent line.

A tangent line is the slope of a graph at 1 particular point of the graph. It gives us the instantaneous rate of change (slope) at a particular point.

200

Given that f(x)=2/x and f'(x)= -2/x², find the equation of the tangent line at x=2.

y=-1/2x+2

300

Is the function f(x)=10x-7 always increasing or decreasing? Explain your answer.

You can only determine if the function is positive or negative with the slope. How do we find the slope? f'(x)=m

400

Find the equation of the tangent line for f(x)=-1/4x² at the point (-1, f(-1)).

y=1/2x-3/4

Differentiable vs Not graph

100

If f is not continuous at x=a then f is nondifferentiable. True or False.

True

200

Discontinuities and sharp corners at x=a are the only criteria that would make f nondifferentiable. True or false.

False

300

Not all forms of sharp corners will make f nondifferentiable. True or False.

False

400

When f has a vertical tangent line at x=a, the blank of the tangent line actually DNE. Fill in the blank.

The slope of the tangent line actually does not exist.