Topic 3 - Final

Question 1

List out the two definitions of the derivative

Answer 1

difference quotient
lim x → a f(x)-f(a)/x-a

limit definition
lim h → 0 f(a+h) - f(a)/h

Question 2

Explain why the following is true?

If f is differentiable everywhere and f(7)=f(12) then the graph of f has a horizontal tangent line between 7 and 12

Answer 2

• If f is differentiable implies continiuty
• Satisifies the conditions of rolle’s theorem, differentiable and continous and then f(a) = f(b) at some point

Can also be shown with MVT
f’(c) = f(b) - f(a)/b - a
since they are equal their difference will equal 0

Question 3

Derivative of:

a^x
logax

Answer 3

1)

a^x * ln(a)

2)

1/xln(a)

A way to remember these is thinking about just differentiating e^x and lnx just scaling to the factor of ln(a)

Question 4

What is the derivative of

f√x

Answer 4

You have to remember chain rule

so you differentiate the root function and multiply by the derivative of the function

f’(√x)
2√x

Question 5

3.2

Two tangent lines to curve of y = x^2 that pass through the point

Answer 5

1. Sketch the graph and label
2. Point we are looking for is a, a^2 (sub in a into the function which is x^2
3. Slope would be f’(a) which is 2a
4. Slope is also the rise/run between 3/4, 0 and a, a^2
5. Set these two equal and solve for a, which is x value of the point of tangent line
6. Now sub in f’(a) with the value from 5 to get the slope of the tangent line at the derivative
7. Sub in y = mx + b format to find b

Question 6

Where is a function not differentiable?

Answer 6

Cusp, corner, vertical asymptote, or endpoint