Theorems CALC

Question 1

Fermats theorem

Answer 1

If f(x) has
- local extrema at c
- f’(c) exists
Then f’(c)=0 because when the derivative is zero it’s a change of direction creating a ‘ceiling’ or ‘floor’

Question 2

Rolle’s theorem

Answer 2

Trying to prove the function has one

• conts on a closed interval [a,b]
• differentiable on open interval (a,b)
• f(a) = f(b)

Prove that the function has a number C with which f’(c)=0
If you can’t prove that then it only has one real root

Question 3

Intermediate value theorem

Answer 3

You need to prove continuity to use it, then you pick two points that prove it crosses the x-axis, one positive and one negative.. you use that to state that it must have at least one real root

Question 4

Median value theorem

Answer 4

More general form of Rolle’s Theorem
You must state that 
- f(x) is conts on [a,b]
- f(x) is differentiable on (a,b)
Then you state there must be a C in (a,b) such that 
              f(b) - f(a)
f’(c) =   —————
                  b - a

Question 5

Squeeze Theorem

Answer 5

• look for trig in a limit
• -1<=trig<=1 for any x!=0
Function >= 0 always
multiply function through 
• -function<=xtrig<=function. 
Fu
Then prove as x->0 the negative or positive xsquared is also equal to 0
Then you state by Squeeze Theorem the limit must be 0 
Because both parts equal zero

Question 6

Both definitions of limits

Answer 6

lim f(x+h) - f(x)
h->0 ———————
h

lim. f(x) - f(a)
x->a ——————
x - a

Question 7

Extreme Value Theorem

Answer 7

If the function is conts domain on a closed interval it has extrema at some number c,d

Question 8

Linear Approximation

Answer 8

Choose some number A close to the number you’re trying to approximate, and make sure A will be easy to calculate in the function

L(x) = f(a) + f’(a)(x-a)

Question 9

L’Hopitals Rule

Answer 9

Use when you have
0 Infinity
lim — or lim ————
0 Infinity

Prove it’s an indeterminate form, then take the derivative of top and bottom(not quotient rule!) and solve limit

Or we have an indeterminate product, where f(x) = 0 and g(x) = infinity… divide g(x) by 1 over f(x) to make it infinity over infinity again

Question 10

Curve Sketching!

Answer 10

Vertical asymptotes 
 - look at where f(x) is undefined
- where it has roots 
Horizontal asymptotes
 - where x-> infinity
 - take limit where x->+-infinity
- number line!! For y value pos or neg 
1st derive test
 - increasing / decreasing
- numberline for incr/decr
 2nd derive test 
 - concavity 
 - numberline for concavity!