Unit 2

Question 1

Every polynomial becomes infinite as x does

Answer 1

becomes positively/negatively infinite, depending only on the sign of the leading coefficient and the degree of the polynomial

Question 2

horizontal asymptote: y=b is a horizontal asymptote of the graph if

Answer 2

lim (x-> ∞) f(x) = b or lim (x->-∞) f(x)

Question 3

THEOREMS ON LIMITS

Question 4

lim kf(x) =

Answer 4

k lim f(x)

Question 5

lim [f(x) + g(x)] =

Answer 5

lim f(x) + lim g(x)

Question 6

lim f(x)g(x) =

Answer 6

(lim f(x))(lim g(x))

Question 7

lim f(x)/g(x) =

Answer 7

lim f(x) / lim g(x) (if lim g(x) does not equal 0)

Question 8

lim (x->c) k =

Answer 8

k

Question 9

lim (x-> ∞) 1/x =

Answer 9

0

Question 10

Limit of a Quotient of Polynomials

Answer 10

the limit of a quotient of two polynomials P(x)/Q(x) , as x approaches a finite value, can be found by directly substituting the value into the polynomials because all polynomials are continuous

Question 11

Rational Function Theorem

Answer 11

states that if a polynomial equation with integer coefficients has a rational solution, that solution can be expressed as a fraction p/q, where p is a factor of the constant term (term without any variable) and q is a factor of the leading coefficient

Question 12

CONTINUOUS FUNCTIONS

Answer 12

• polynomials
• rational functions P(x)/Q(x) except where Q(x)=0
• absolute value functions
• trigonometric, inverse trigonometric, exponential, logarithmic functions

Question 13

Extreme Value Theorem

Answer 13

if f is continuous on the closed interval [a, b], then f attains a minimum value and a maximum value somewhere in that interval

Question 14

Intermediate Value Theorem

Answer 14

if f is continuous on the closed interval [a, b], and M is a number such that f(a) </= M </= f(b), then there is at least one value, c, in the interval [a, b], such that f(c) = M

Question 15

derivatives do not exist where _____ exist

Answer 15

asymptotes, holes, vertical tangents, and sharp points

Question 16

instantaneous rate

Answer 16

derivative at the given point

Question 17

average rate

Answer 17

slope of the endpoints (dy/dx)

Question 18

position

Answer 18

feet: s(t) = -16t^2 + Vot + So
meters: s(t) = -4.9t^2 + Vot + So

• Vot is initial velocity
• So is initial height

Question 19

velocity

Answer 19

derivative of position, integral of acceleration

Question 20

acceleration

Answer 20

derivative of acceleration

Question 21

quotient rule

Answer 21

low dHi - Hi dlow over the bottom squared and away we go

Question 22

DERIVATIVES

Question 23

d/dx (tanx) =

Answer 23

sec^2 x

Question 24

d/dx (cotx) =

Answer 24

-csc^2 x

Question 25

d/dx (secx) =

Answer 25

secxtanx

Question 26

d/dx (cscx)

Answer 26

-cscxcotx

Question 27

csc^2 x - 1 =

Answer 27

cot^2 x

Question 28

d/dx ln(x) =

Answer 28

1/x

Question 29

d/dx [a^x] =

Answer 29

a^x ln(a)

Question 30

derivative of an inverse

Answer 30

f(x, y) -> f'(y, x)

Question 31

INVERSE TRIG DERIVATIVES

Question 32

d/du arcsin(u) =

Answer 32

1/(1-u^2)^1/2 u' - arccos(u) is negative

Question 33

d/du arctan(u) =

Answer 33

1 / 1+u^2 u' - arccot(u) is negative

Question 34

d/du arcsec(u) =

Answer 34

1 / |u|(u^2 - 1)^1/2 u' - arccsc(u) is negative

Question 35

RELATED RATES

Answer 35

- PAY ATTENTION to what the rates mean in respect to time and take the derivative of a formula to find whatever it is to be found - ex. area/volume formulas, pythagorean theorem a^2 + b^2 = c^2

Question 36

Volume sphere

Answer 36

V = 4/3 πr^3

Question 37

volume cone

Answer 37

V = 1/3 πr^2 h