chapter 4 (applications of derivatives)

Question 1

optimization problem steps

Answer 1

1. identify 2 formulas
2. solve for one of them and plug it into the other.
3. take the derivative of the new function
4. find the critical points
5. this is the maximum or minimum

Question 2

how to approximate F(x) given a

Answer 2

1. plug in a to find its y coordinate
2. take the derivative of the equation and plug in a to find the tangent line slope
3. use the point slope formula and plug in x. this should allow you to get to y

Question 3

rolles theorem

Answer 3

if the function is continuous and differential and f(a) and f(b) are equal then it is guaranteed there is a point with the slope of 0

Question 4

finding guaranteed points with rolls theorem

Answer 4

1. determine if it will work
2. derive the equation and set equal to zero.
   ta da

Question 5

mean value thorem

Answer 5

if the function is continuous and differentiable from a to b then it is certain that there exists a point within the interval a to b that has a slope equal to the average slope

Question 6

find values with mean value theorem

Answer 6

1. see if it will work
2. find the mean slope( y2-y1/x2-x1)
3. derive and set equal to

Question 7

lopitals o/o and inf/inf

Answer 7

1. derive the top and bottom separately

2. take the limit of the top over the limit of the bottom

Question 8

lopitals 0*inf

Answer 8

1. make it into a fraction by placing one of the sides on the bottom to the -1 power.
2. then treat like a normal lopitals

Question 9

lopitals 0^inf inf^0 etc

Answer 9

1. take the ln of BOTH sides
2. use log rules to make it multiplication
3. use the multiplication to find the limit.
4. undo the log on the other side by taking e^ what you get

Question 10

power rule for anti-derivatives

Answer 10

(x^p+1)/p+1

Question 11

d/dx(sin ax)

Answer 11

acos ax

Question 12

d/dx cos ax

Answer 12

-asin ax

Question 13

d/dx tan ax

Answer 13

asec^2(ax)

Question 14

d/dx cot ax

Answer 14

-acsc^2 ax

Question 15

d/dx sec ax

Answer 15

a sec(ax) tan(ax)

Question 16

d/dx csc ax

Answer 16

-a csc(ax) cot(ax)

Question 17

int cos ax

Answer 17

(1/a)sin ax +C

Question 18

int sin ax

Answer 18

-(1/a)cos ax +C

Question 19

int sec^2 ax

Answer 19

(1/a)tan ax +C

Question 20

int csc^2 ax

Answer 20

-(1/a)cot ax +C

Question 21

int sec(ax) tan(ax)

Answer 21

(1/a)sec ax +C

Question 22

int csc(ax) cot(ax)

Answer 22

-(1/a) csc ax +C

Question 23

d/dx e^ax

Answer 23

ae^ax +C

Question 24

d/dx b^x

Answer 24

(b^x)lnb +C

Question 25

d/dx lnx

Answer 25

1/x +C

Question 26

d/dx sin^-1 (x/a)

Answer 26

1/(sqrt(a^2-x^2)) +C

Question 27

d/dx cos^-1 (x/a)

Answer 27

-1/(sqrt(a^2-x^2)) +C

Question 28

d/dx tan^-1 (x/a)

Answer 28

a/(a^2+x^2) +C

Question 29

d/dx cot^-1 (x/a)

Answer 29

-a/(a^2+x^2) +C

Question 30

d/dx sec^-1 (x/a)

Answer 30

a/(xsqrt(x^2-a^2)) +C

Question 31

d/dx csc^-1 (x/a)

Answer 31

-a/(xsqrt(x^2-a^2)) +C

Question 32

int e^ax

Answer 32

(1/a)e^ax +C

Question 33

int (b^x)

Answer 33

(1/lnb)b^x +C

Question 34

int (1/x)

Answer 34

ln x +C

Question 35

int 1/(sqrt(a^2-x^2))

Answer 35

sin^-1 (x/a) +C

Question 36

int 1/(a^2+x^2)

Answer 36

(1/a) tan^-1 (x/a) +C

Question 37

int 1/(xsqrt(x^2-a^2))

Answer 37

(1/a) sec^-1 (x/a) +C

Question 38

how to find the integral equation at a point

Answer 38

solve for the indefinite integral and then set equal to the point to find c

Question 39

how to find the critical point of a function

Answer 39

find teh derivative and set equal to zero

Question 40

for a local extrema to exist what must the derivative at that point be

Answer 40

zero

Question 41

a function that is continuous from a to b has what

Answer 41

a maximum and minimum on that interval

Question 42

how to tell if a function is increasing or decreasing.

Answer 42

take the derivative and identify your interval.
if the function is negative in that interval then it is decreasing
if positive then it is increasing

Question 43

if f’ goes from negative to positive at c

Answer 43

local minimum

Question 44

if f’ goes from positive to negative at c

Answer 44

local maximum

Question 45

inflection points

Answer 45

when the function switches fro concave up to concave down. found by setting the second derivative to 0

Question 46

graphing steps

Answer 46

find both derivatives. 
1. window
2. critical points and values
3. inflection point and values
4. x and y intercepts
5 asymptotes if any

Question 47

if f has a slope of 0 what is the f’ graph doing

Answer 47

passing through 0

Question 48

when the slope of the f’’ graph passes through zero, what is this

Answer 48

an inflection point