Calc 1 Midterm Review :P

Limit Logic

100

Define what it means for a limit to exist at x=a.

The left-hand and right-hand limits both exist and are equal: lim⁡x→a−f(x)=lim⁡x→a+ f(x)

200

Evaluate lim⁡_x→3 (x²-9)/((x-3)

6

300

A limit gives 0/0. What does that mean?

It's indeterminate, you must simplify or rationalize before evaluating

400

Evaluate lim_x→0 sin(3x)/x

3

500

f(x)={x²              x<2

         3x-2          x≥2

Is f continuous at x=2?

Yes, both sides equal 4

Algebra

100

Simplify (1/(x+h) - 1/x)/h

-1/(x(x+h)

200

Simplify lim_x→4 √(x-2)/(x-4)

1/4

300

Find the derivative of f(x) = 1/x using the limit definition

f'(x) = -1/x²

400

Rationalize the numerator (√(x+1) - √x )/h

1/(√(x+1) + √x )

500

Simplify 1/(1/x + 1/y)

xy/(x+y)

Trig

100

What is sin(0)?

0

200

Evaluate lim_x→0 sin(x)/x

1

300

Evaluate lim_x→0 (1-cos(x))/x²

1/2

400

What is the d/dx sin(x), d/dx cos(x), d/dx tan(x)

cos(x), -sin(x), sec²(x)

500

What’s sin⁡(π/6), cos⁡(π/6), and tan(π/6)?

1/2, √3/2, 1/√3

Derivatives

100

State the limit definition of the derivative

f'(x) = lim_h→0 (f(x+h)-f(x))/h

200

Interpret f'(a) graphically

It's the slope of the tangent line at x=a

300

Find f'(x) for f(x) = (x²+3x)/(x+1) using standard derivative rules

(x²+2x+3)/(x+1)²

400

Use the limit definition of the derivative to find f'(x) for f(x) = √(x+2)

1/(2√(x+2))

500

f(x) = 1/(x+1). Find f'(x) using any method, then find the equation of the tangent line at (1,1/2)

f'(x) = -1/(x+1)²

y= (-1/4)x + 3/4

Piecewise and Continuity

100

Define continuity at x=a

f(a) exists, lim_⁡x→af(x) exists, left and right hand limits are equal

200

Determine if f(x) = { x+2,   x<1

                                          x²,      x≥1

is continuous at x=1

Yes, both sides equal 2

300

Find the value of k that makes 

f(x) =  {x²,       x<2

            kx-2,    x≥2

left and right hand limits must equal

-->4=2k-2

      --> k=3

400

Let f(x) = {x²,       x<1

                 2x,       x≥1

Determine whether f(x) is continuous and differentiable at x=1

1. must be continuous at x=1

2. Its left and right derivatives must be equal at x=1

--> left and right hand don't match so        not continuous

--> if a fan is not continuous, it can't be differentiable (but let's still check)

500

f(x)={ x²,            x<1

           2x-1,        x≥1

1. Is f(x) continuous at x=1?

2. Is f(x) differentiable at x=1?

3. If differentiable, find the equation of the tangent line at (1,1)

Continuous and differentiable

y=2x-1