Differentiation

Question 1

f is differentiable at c if

Answer 1

the limit lim(x→c) f(x)-f(c)/x-c exists and is finite

OR

lim (h→0) f(c+h)-f(c)/h = k

Question 2

I is an interval containing c, f differentiable at c iff for every (x_n)→c, x_n not equal to c,

Answer 2

f(x_n)-f(c)/

x_n-c

Question 3

If f is differentiable at c in I,

Answer 3

then f is continuous at c

Question 4

Arithmetic of derivatives: f and g differentiable, then

Answer 4

(kf)’ = k * f’

(f+g)’ = f’ + g’

(fg)’ = fg’ + gf’

(f/g)’ = (gf’-fg’)/(g²)

Question 5

Chain Rule

Answer 5

(g o f)’ = g’(f) * f’

Question 6

f differentiable on (a, b) and if assumes max or min at c in (a,b),

Answer 6

then f’(c)=0

Question 7

Rolle’s Theorem: f continuous and differentiable on (a, b), f(a)=f(b)

Answer 7

there is at least one c such that f’(c)=0

Question 8

Mean Value Theorem: f continuous [a,b] and differentiable on (a,b), then there is c in (a,b)

Answer 8

f’(c) = (f(b)-f(a)) / (b-a)

Question 9

f continuous on [a,b], differentiable (a,b). If f’=0 for all x in (a,b),

Answer 9

then f is constant on [a, b]

Question 10

strictly increasing/decreasing, if f’(x)< or > 0 for all x in an interval, then

Answer 10

f’<0, decreasing

f’>0 increasing