Continuities and Derivatives

Question 1

Only discontinuous if

• There is no _ (asymptote)
• _ (LHL!=RHL)
• lim(x->a) != _

Answer 1

Y value, DNE, f(a)

Question 2

Three steps to prove _

• LHL
• RHL
• LHL=RHL

Answer 2

Continuous

Question 3

Intermediate Value Theorem

• Plug in [] in f(x) to get ()
• By the intermediate value theorem, since 0 is between (y range) there is a root on the interval [x range]

Answer 3

X range, y range

Question 4

Derivatives

• Plug in f’(x) = lim(h->0) [f(x+h) – f(x)] / h into _
• Simplify
• Cancel h
• Plug in _ for h

Answer 4

f(x), 0

Question 5

Solve for f’(a)

• Plug in _ in f’(x) for x
• Plug in f’(x) in _
• Simplify
• Cancel h
• Plug in 0 for h

Answer 5

a, f(x)

Question 6

Alternate _

• Plug in a into f’(a) = lim(x->a) [f(x) – f(a)] / (x-a)
• Cancel

Answer 6

f’(a)

Question 7

Graphing slope of f’(x)

• _
• Plot + and –
• Plot _
• Graph slope

Answer 7

Arrows, 0

Question 8

Prove continuous on (-infinity, infinity)

• lim(x->n+) = _
• Solve

Answer 8

lim(x->n-)