5 Page Memory Quiz

Question 1

1. formal definition of a derivative

Answer 1

if y=f(x) then f’(x)=limit as h ➡ 0 f(x+h)-f(x)/h

Question 2

2. where are critical values located?

Answer 2

x values where the second derivative=0 or does not exist

Question 3

3. where are inflection points located?

Answer 3

x values where the 2nd derivative=0 or does not exist and changes from ➕ to ➖

Question 4

4. increasing and decreasing is related to what derivative?

Answer 4

1st

Question 5

5. CU and CD is related to what derivative?

Answer 5

2nd

Question 6

6. diagram to determine relative extrema by the 1st derivative test

Answer 6

if f’(x)=0 at x=a and
f’(x) __➕__|__➖__ (rel max at x=a)
OR
f’(x) __➖__|__➕__ (rel min at x=a)

Question 7

7. relative extrema by the second derivative test

Answer 7

if f’(a)=0 at x=a and f’‘(a)>0 there is a rel min at x=a and if f”(a)<0 there is a rel max at x=a

Question 8

8. 3 conditions for a function to be contiunuos at a point

Answer 8

1. f(c) is defined
2. lim x➡c f(x) exists
3. lime x➡c f(x)=f(c)

Question 9

9. difference between essential and removable discontinuity

Answer 9

essential: occurs where there is a vertical asymptote or a jump
removable: occurs where the function has a hole

Question 10

10. where does a derivative not exist?

Answer 10

AT X=A: hole, vertical asymptote, jump, corner point, vertical tangent, cusp

Question 11

11. difference between average rate if change and instantaneous rate of change

Answer 11

ave rate is the slope of the secant line and instant is the slope of the tangent line

Question 12

12. f’(x)=(x+2)(x-1)(x-3) writing part

Answer 12

f INC (a,c)u(e,➕➰) , DEC (➖➰,a)u(c,e) , max at x=c min at x=a,e
f CU (➖➰,b)u(d,➕➰) , CD (b,d) , IP at x=b,d

Question 13

13. where are zeros, discontinuities, vertical asymptotes, holes, and horizontal asymptotes?

Answer 13

zeros: numerator=0, disc.: denominator=0, vert. asymp.: non-cancelled factors in denominator=0, holes: cancelled factors=0, horz. asymp.: lim x➡➕➰ f(x)

Question 14

14. derivative of an inverse

Answer 14

y=f(x) ➡ inv. x=f(y) ➡ d/dx[x]=d/dx[f(y)] ➡ 1=f’(y)d/dx ➡ dy/dx=1/f’(y)

Question 15

15. point-slope form of equation of a tangent line

Answer 15

y-Y1=f’(✖1)(✖-✖1)

Question 16

16. calculating local linearization (tan line approx)

Answer 16

f(✖o +

Question 17

17. when do you use logarithmic differentiation? give an example.

Answer 17

when you have a variable function raised to a power that is also a variable function. y=x^x^2 ➡ lny=x^2lnx ➡ d/dx[lny]=d/dx[x^2lnx] ➡ 1/y•dy/dx=x^2• 1/x+lnx•2x ➡ dy/dx=(x + 2xlnx)•x^x^2

Question 18

18. what is a reimann sum approx?

Answer 18

approx of the area under a curve. the area is approx by adding a finite # of rectangles whose height is determined by using the left, right, or midpoint of each rectangle. width of each rect. is b-a/n and the height is f(✖k) where ✖k is the L, R, or M value of each rect.

Question 19

18. reimann sum example

Answer 19

1/50 [✔1/50 + ✔2/50 + ✔3/50 + … ✔50/50]=⤴0,1 ✔xdx
width, height, interval [0,1]
right endpoint reimann sum

Question 20

1. mean value theroem thereom stated

Answer 20

if f is differentiable on (a,b) and continuous on [a,b] then there is at least one number c in (a,b) such that f’(c)=f(b)-f(a)/b-a

Question 21

1. mean value thereom simply stated

Answer 21

the MVT is where the slope of the tangent equals the slope of the secant

Question 22

2. rolles thereom

Answer 22

if f is differentiable on (a,b) and continuous ob [a,b] and f(a)=f(b)=0 then there is atleast one number c in (a,b) such that f’(c)=0

Question 23

3. intermediate value thereom

Answer 23

if f is continuous on a closed interval [a,b] and k is any number between f(a) and f(b) inclusive then there is atleast one number c in the interval [a,b] such that f(c)=k

Question 24

4. intermediate value thereom applied to finding roots

Answer 24

if f is continuous on [a,b] and f(a)>0 and f(b)0 then there is atleast one zero of f on (a,b)

Question 25

5. extreme value thereom

Answer 25

if a function f is continuous on a finite interval [a,b] then f has both an abs max and an abs min. extrema will occur at critical values or at an endpoint of [a,b]

Question 26

6. l'hopitals rule

Answer 26

it is used to find the limit of a rational function whose limit is in the form of lim x➡a f(x) =0/0 , the rule is: lim x➡a f(x)/g(x)=lim x➡a f'(x)/g'(x)

Question 27

7. mean value thereom for integrals

Answer 27

⤴a,b f(x)dx=f(c)(b-a) | ⤴a,b f(x)dx)=height(f(c))•width((b-a)

Question 28

8. average value of a function

Answer 28

fave=1/b-a⤴a,b f(x)dx | height fave or (f(c))=1/width((b-a))•area(⤴a,b f(x)dx)

Question 29

9. fundamental thereom of calc part 2 with lower limit a constant and upper limit g(x)

Answer 29

d/dx ⤴a,g(x) f(t)dt= f(g(x))•g'(x)

Question 30

10. fundamental thereom of calc part 2 with lower limit as h(x) and upper as g(x)

Answer 30

d/dx ⤴h(x),g(x) f(t)dt= f(g(x))•g'(x) - f(h(x))•h'(x)

Question 31

1. explicit formula for exponential growth

Answer 31

y=ce^kt

Question 32

2. differential equation for exponential growth

Answer 32

dy/dt=ky or y'=ky

Question 33

3. area between curves

Answer 33

a=⤴a,b [f(x)-g(x)]dx

Question 34

4. volume based on cross sections

Answer 34

v=⤴a,b a(x)dx

Question 35

5. volume formed by discs

Answer 35

v=♊•⤴a,b[f(x)]^2

Question 36

6. volume formed by washers

Answer 36

v=♊•⤴a,b [(f(x))^2-(g(x))^2]dx