differentiation

Question 1

differentiable function

Answer 1

A function f is differentiable at a if f’(a) exists. It is differentiable on an open interval (a,b) or (a,∞) or (-∞, b) or (-∞,∞) if it is differentiable at every number in the interval.

Question 2

continuity & differentiability

Answer 2

If f is differentiable at a number a, then f is continuous at that number a.

f is differentiable → f is continuous
does not work backwards, no CDs

Question 3

f(x) = c
f’(x)

Answer 3

= 0

Question 4

f(x) = x
f’(x)

Answer 4

= 1

Question 5

power rule

Answer 5

if n is any real number, then

d/dx (x^n) = nx^n-1

Question 6

if c is a constant and f is a differentiable function then d/dx[cf(x)] =

Answer 6

c (d/dx f(x))

Question 7

product rule

Answer 7

[f(x)g(x)]’ = f(x)g’(x) + f’(x)g(x)

Question 8

f(x) = a^x
f’(x) =

Answer 8

a^x lna

Question 9

defn of number e

Answer 9

e is the number such that f’(0)
=1

the function f(x) = e^x is the one whose tangent line at (0,1) has a slope f’(0) that is exactly 1

Question 10

deriv of e^x

Answer 10

e^x

Question 11

f(x) = 3e^x
f’(x)
f’‘(x)

Answer 11

= 3e^x
= 3e^x

Question 12

f’(sinx)

Answer 12

cosx

Question 13

f’(cosx)

Answer 13

-sin(x)

Question 14

f’(tanx)

Answer 14

sec^2 (x)

Question 15

f’(secx)

Answer 15

sec(x)tan(x)

Question 16

f’(cotx)

Answer 16

= csc^2(x)

Question 17

chain rule
F = f ∘g

Answer 17

f’ (g(x)) g’(x)

dy x du

Question 18

d/dx of logbx

Answer 18

1/xlnb

Question 19

d/dx lnx

Answer 19

1/x

Question 20

deriv of lng(x)

Answer 20

= g’(x)/g(x)

Question 21

y = logbx alt

Answer 21

b^y = x

Question 22

extreme value theorem

Answer 22

For any continuous function defined on a closed interval [a,b], there exists a global max f(c) at some point cE[a,b], and there exists a global ,min f(d) at some point dE[a,b]

Question 23

fermat’s theorem

Answer 23

If f has a local maximum or minimum at x=c AND if f’(c) exists , then f’(c) =0

doesnt work reverse

Question 24

critical number

Answer 24

A critical number is a number c in the domain of a function where f’(c) =0 or f’(c) DNE

Question 25

theorem to find extreme

Answer 25

If f has a local max or min at a point x=c then f has a critical number @ x=c

Question 26

find global max and min

Answer 26

1. Find critical numbers of f on (a,b) 2. Check the numbers at the endpoints f(a) and f(b) 3. Compare values from step 1 and 2 and find minimum & maximum

Question 27

mean value theorem collary

Answer 27

If f'(x) = g'(x) for all x in the interval (a,b) then f - g is constant on (a,b), that is f(x) = g(x) + c, where c is a constant if they differ by a constant (addition of constant) then gradients are the same

Question 28

increasing fn

Answer 28

if f'(x) >0 on an interval then f is increasing on that interval

Question 29

decreasing function

Answer 29

if f'(x) <0 on an interval then f is decreasing on that interval

Question 30

find extrema using 1st deriv

Answer 30

- find 1st deriv -find CVs -Sub sample points w/in interval into derivative formula to see whether gradient is + or - and subsequently where graph is increasing or decreasing to see where local max and mins exist -Then sub CVs and boundary values into f(x) -Now compare values

Question 31

second derivative

Answer 31

If the tangents of a graph on some interval I lie below the graph, then we say the graph is concave up. If the tangents of a graph lie above the graph on I, then we say the graph is concave down on I

Question 32

concave up

Answer 32

f"(x) > 0 U

Question 33

concave down

Answer 33

f"(x) < 0

Question 34

inflection point

Answer 34

A point P on a continuous curve y=f(x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or concave downward to concave upward at P. Here f'=0

Question 35

second derivative test

Answer 35

if f'(c) = 0 and f"(c) > 0, f has a local min at c if f'(c) = 0 and f"(c)< 0, f has a local max at c

Question 36

find extrema usimg 2nd deriv test

Answer 36

no endpoints Find first derivative, CVs, and second derivative Sub CVs into first and second derivatives

Question 37

2 indeterminate forms

Answer 37

If thr