Quater 3 Final

Question 1

Instantaneous rate of change other names
ex. with 3
units

Answer 1

limit (unofficial)
derivative
see what f(2.9) is then f(3.1) what number is in the middle is the IRC of 3
yunits/x units

Question 2

a. what does E and 8 go with?
b. which 8 is better bigger of smaller?

Answer 2

a. y and x
b. smaller

Question 3

The Limit Theorems
a. Constant Times
b. Identity
c. Constant

Answer 3

a. the limit of a constant times a function equals the constant times the limit
b. lim(x->c)x = c
c. if f(x) = k and k is constant then lim(x->c)f(x) = k

Question 4

Continuity on an interval

Answer 4

function f is continuous on an interval of x-values iff its continuous at each value of x in that interval
At the endpoints of a closed interval, only one-sided limits need to equal the function value

Question 5

a. IVT theorem definition
b. IVT is not what type of theorem and what does that mean?

Answer 5

a. If function f is continuous for all x in the closed interval [a, b], and y is a number between (NOT EQUAL TO) f(a) and f(b), then there is a number x = c in
(a, b) for which f(c) = y
b. IVT is not an iff theorem, which means the conclusion can be true even if IVT does not apply to a function

Question 6

Definition of derivative at a point

Answer 6

f ‘(c) = lim x->c (f(x)-f(c)/x-c)

Question 7

Derivative of a function (equation)

Answer 7

f’(c) = lim h->0 ((f(x+h) - f(x)) / h)

Question 8

a. How do you tell if an object is increasing or decreasing?
b. How do you tell if the object is speeding up or slowing down

Answer 8

a. if v is positive its increasing, if v is negative its decreasing
b. if the signs (positive or negative) are the same for acceleration and velocity then the object is speeding up. If the signs are not the same then the object is slowing down

Question 9

General facts:
a. log(subscriptb)y = x. = ?
b. lnx =
c. 1/x^3 =
d. lne =
e. lne^a =
f. log(subscriptb)(b^x) =
g. log(subsciptb)b =
h. b^log(subscriptb)x =
i. log(b) 0 =
j. a/oo =
k. oo/a =

Answer 9

a. b^x=y
b. log(e)x
c. x^-3
d. 1
e. a
f. xlog(subscriptb)b=x
g. 1
h. x (also applies to e^lnx =x)
i. indeterminate (the limit would be infinity)
j. 0 (limit) a is a smallish number
k. oo(limit) a is any smallish number

Question 10

Quotient Rule for derivatives

Answer 10

y’ = u’v - uv’/v^2

Question 11

What is the derivative of (sr stands for square root):
a. sin^-1x and for cos add a ____
b. tan^-1x and for cot add a ____
c. sec^-1x, and for csc add a ____
d. Remember x is the _________ and remember you still have to apply ________

Answer 11

a. 1/sr(1-x^2), add - for cos
b. 1/1+x^2, add - for cot
c. 1/|x|sr(x^2 -1), add - for csc
d. argument, chain rule

Question 12

Fun extra things to know! ((sr is squarerootof)
a. cos^2x + sin^2x =
b. tan^2x + 1 =
c. cot^2x+1=
d. 30 - 60 -90 triangle sides and corresponding pi stuff
e. 45-45-90 triangle sides and corresponding stuff
f. how do you know if a pi something is positive or negative
g. The derivative of a constant is ________ a fun way to check if its a constant is _____

Answer 12

a. 1
b. sec^2x
c. csc^2x
d. across from 90 is 2, across from 30 (pi/6) is x and across from 60 (pi/3) is sr 3
e. across from both 45s (pi/4) is 1 and across from 90 is sr 2
f. cast, c is in the 4th quadrant and a is in the 1st and so on
g. 0, if it has a variable (it does not a constant, it doesn’t it is a constant)

Question 13

What is the equations for a volume of a cylinder and when do you use it?

Answer 13

v=pir^2h
when your rotating around something

Question 14

A region in quad 1 is bound by y = 4-x^2 set up equation for volume and do not solve for:
a. rotate around y axis
b. rotate around x = -3
c. rotate around x = 3
d. rotate around x axis
e. rotate around y = -5
f. rotate around y=5

Answer 14

a. int0-4 pi(sqr4-y)^2dy
b. int0-4 pi((sq4-y)^2 + 3^2) dy - int0-4 pi 3^2 dy
c. int0-4 pi 3^2 dy - int0-4 pi(3^2 - (sq4-y)^2) dy
d. int 0-2 pi(4-x^2)^2 dx
e. int0-2 pi (4-x^2)^2 + 5^2 dx - int0-2 5^2 dx
f. int0-2 pi5^2dx - int0-2(5^2-(4-x^2)^2)dx

Question 15

Let r be the region bounded by the graph of y1 = 6e^-0.2x and y2=sqx and by the vert lines x = 1 and x=4 find the volume when x is rotated around the x axis,

Answer 15

int1-4 pi y1^2dx - int1-4 pi y2^2 dx

Question 15

a. sin pi/2
b. sin 0
c. cos 0
d. tan 0
e. sec 0

Answer 15

a. 1
b. 0
c. 1
d. 0
e. 1

Question 16

Fundamental theorem of calculus

Answer 16

if g(x)=inta->x f(t) dt where a is a constant and f is cont by a then g’(x) = f(x)

Question 17

Derivatives:
a. lnx =
b.int 1/u du =
c. logbx =
d.b^x =
e.int b^u du =
f. intlnx dx=
g. int 1->x 1/t dt =

Answer 17

a.int1->x 1/t dt
b.ln|u| + c
c. logax/logab = lnx/lnb
d.b^xlnb
e.b^u/lnb + c
f. lnx-x+C
g. lnx

Question 18

l’Hospital’s Rule

Answer 18

If f(x) = g(x)/h(x) and if limx->c g(x)=limx->ch(x)=0
then limx->cf(x) =
limx->c g’(x)/h’(x)

Question 19

a. If f’(x) >0 then
b. If f’(x) is < 0 then
c. relative max occurs when
d. relative min occurs when

Answer 19

a. f(x) is increasing
b. f(x) is decreasing
c. f’ changes from + to -
d. f’ changes from - to +

Question 20

a. f’‘>0 means
b. f’‘<0 means
c. f’’ = 0 means

Answer 20

a. relative min
b. relative max
c. test fails

Question 21

For finding absolute/global points what do you do?

Answer 21

find y values of each critical point and end points in the domain
absolute max is largest y value, absolute min is smallest y value

Question 22

a. If f’‘(x) > 0 then f(x) is
b. If f’‘(x) < 0 then f(x) is
c. using f’ how do you know if theres an inflection point
d. using f’’ how do you know if theres an inflection point

Answer 22

a. concave up
b. concave down
c. if it changes from increasing to decreasing or decreasing to increasing there is an inflection point
d. if f’’ changes from pos to neg or neg to pos that means a point of inflection

Question 23

Continuity at a point

Answer 23

function f is continuous at x = c iff
1. f(c) exists
2. lim(x->c) f(x) exists
3. lim(x->c) f(x) = f(c)

Question 24

Trig Identities (t is theta)
a. for even odd identities which ones are even and which ones are odd (they look like: sin(-t)= ___
b. sin2t =
c. sin^2t=
d. cos^2t=
e. tan^2t=

Answer 24

a. everything except cos and sec are negative
b. 2sintcost
c. (1-cos2t)/2
d. (1+cos2t)/2
e. (1-cos2t)/(1+cos2t)

Question 25

a. Riemann Sums
b. Rolls
c. MVT

Answer 25

a. rectangles!
b. f is diff for an x in (a, b) and f is cont for all x in [a, b] and f(a) =f(b) = 0then x=c in (a,b) so f’(x) =0
c. f is diff for all x in (a, b) and f is cont for all x in [a, b] then there is at least one x=c in (a, b) so that f’(c) = f(b)-f(a)/ b-a

Question 26

a. Integration by parts
b. distance and displacement
c. average velocity

Answer 26

a. intudv = uv- intvdu
b. displacement: int a->b velocity dt
distance: inta->b |velocity|dt
c. (inta->b v(t) dt)/b-a