All Calculus Units

Question 1

slope/intercept/average rate of change for linear equation

Answer 1

y = mx + b

Question 2

slope formula

Answer 2

m = (y_2 - y_1) / (x_2 - x_1)

Question 3

standard form for linear equation

Answer 3

y = ax + b

Question 4

increasing graph slope =

Answer 4

increasing graph point

Question 5

decreasing graph slope =

Answer 5

decreasing graph point

Question 6

no change graph slope =

Answer 6

0, max/min point, vertex point

Question 7

undefined graph slope =

Answer 7

undefined

Question 8

equation of line

Answer 8

y = m (x-x_1) + y_1

Question 9

a parabola that is negative slope -> positive slope opens

Answer 9

upwards, local minimum point

Question 10

a parabola that is positive slope -> negative slope opens

Answer 10

downwards, local maximum point

Question 11

vertex form

Answer 11

y = a (x-p)^2 + q

Question 12

vertex form steps

Answer 12

1. factor coefficient from a & b
   y = -2(x^2-6x)-23
2. take result ‘b’ coefficient, divide by 2 and square result. Take squared result and add and subtract to brackets
   y = -2(x^2-6x+9-9)-23
3. factor first three terms
   y = -2[(x-3)^2-9]-23
4. redistribute factored variable
   y = -2(x-3)^2 +18-23
5. simplify
   y = -2(x-3)^2 -5

vertex = (+3, -5)

Question 13

quadratic formula

Answer 13

• b +/- root (b^2 -4ac) / 2a

* if quadratic equation -> ax^2 + bx - c = 0

Question 14

factor theorem

Answer 14

if a numbe entered into an equation results in zero it is a factor.

Question 15

what is used to find other equation factors, after determing 1 factor?

Answer 15

long division or synthetic division

Question 16

method of substitution

Answer 16

substituting a linear equation in for a polynomial’s y value to determine what point goes through both lines

Question 17

secant line

Answer 17

straight line that intersects the curve at 2 points

Question 18

tangent line

Answer 18

straight line that touches the curve at one point

Question 19

secant method

Answer 19

determing the instantaneous slope of a curve by calcualting the slope of a point very close to the curve from a tangent line.

(y_2 - Cy+) / (x_2 -(+0.1x_2) = positive slope
(y_2 - Cy-) / (x_2 -(-0.1x_2) = negative slope

Question 20

horizontal limits

Answer 20

lim x-> ∞ (1/x) = 0

lim x-> -∞ (1/x) = 0

Question 21

vertical limits

Answer 21

lim x-> 0^+ (1/x) = ∞

lim x-> 0^- (1/x) = -∞

Question 22

does a limit always exist?

Answer 22

not if it isn’t the same from the right and left
+∞ = -∞
insert x-value into function, result = limit*
*may need to factor or difference of squares
[g(x) = (rootx) -5 / x-25] ->
[g(x) = {(rootx) -5}{(rootx) +5} / {(rootx) -5}{(rootx) +5}] -> g(x) = 1/(root x) + 5

Question 23

limit formula

Answer 23

lim   f(a+h) - f(a) / h
h-> 0

Question 24

extrema

Answer 24

max/min point on a parabola

Question 25

global maximum

Answer 25

aka absolute maximum

Question 26

local maximum

Answer 26

aka critical point aka stationary point

Question 27

general formula for the slope of any tangent line along the curve

Answer 27

``` f(a) = 4a^2 f(a+h) = 4(a+h)^2 -> 4a^2 + 8ah + 4h^2 ``` lim [(4a^2 + 8ah + 4h^2) - (4a^2)]/(a+h) - (a) h->0 = 8a

Question 28

derivative

Answer 28

slope of tangent line at specific point aka instantaneous rate of change f(x) = 2x^2 + 1 -> f'(x) = 4x

Question 29

polynomial function

Answer 29

``` f(x) = a_n x^n + a_n-1 x^n-1 + a_2 x^2 + a_0* *n = natural number a= constants ```

Question 30

degree

Answer 30

maximum number of directions, maximum number -1 = maximum number of turning points, determined by finite differences

Question 31

finite differences

Answer 31

x | y | 1st dif. | 2nd dif. | 0 | 1 | 2-1 = 1 | 7-1=6 | 1 | 2 | 9-2=7 | 13-7=6 | 2 | 9 | 22-9=13 | 19-13=6 | 3 | 22 | 41-22=19 | 4 | 41 |

Question 32

method of first principles

Answer 32

f(x) = 2x^2 + 1 ``` lim f(a+h) - f(a) / h h-> 0 ``` lim (2x^2 + 4xh + 2h^2 +1) - (2x^2 +1) / h h-> 0 = f'(x) = 4x

Question 33

power rule

Answer 33

f(x) = x^n , n ≠ 0 -> f'(x) = nx^(n-1)

Question 34

constant rule

Answer 34

f(x) = c , c = real # -> f'(x) = 0

Question 35

constant multiple rule

Answer 35

f(x) = kg(x) , k = real # -> f'(x) = kg'(x)

Question 36

sum rule

Answer 36

F(x) = f(x) + g(x) -> F'(x) = f'(x) + g'(x)

Question 37

difference rule

Answer 37

F(x) = f(x) - g(x) -> F'(x) = f'(x) - g'(x)

Question 38

product rule

Answer 38

F(x) = f(x)g(x) -> F'(x) = f'(x)g(x) + f(x)g'(x)

Question 39

chain rule

Answer 39

F(x) = (f(x))^n -> F'(x) = nf'(x)f(x)^(n-1)

Question 40

quotient rule

Answer 40

``` F(x) = f(x) / g(x) -> F'(x) = [f'(x)g(x) - f(x)g'(x)] /[g(x)]^2 F'(x) = (LodeeHI-HIdeeLO)/LOxLO ```

Question 41

curve sketching steps

Answer 41

1. find original function derivative 2. substitute '0' to find min/max 3. determine if min or max (left of number = -, min, left of number = +, max)

Question 42

tangent point equation steps

Answer 42

1. find original function derivative 2. substitute tangent point for x-value 3. substitute tangent slope into original function 4. use equation formula (y= m(x-x_1) + y_1

Question 43

polynomial vs. exponential equations

Answer 43

polynomial; y= x^a | exponential; y= a^x (a > 0 ≠ 1)

Question 44

decay vs. growth exponential equations

Answer 44

both have; no x-intercepts, horizontal asymptote at y=0, y-intercept at (0,1), no max/min points, domain {x element of all real numbers}, range {y element of all real numbers, y>0} decay; as x approaches positive infinity, y decreases slower growth; as x approaches negative infinity, y increases faster

Question 45

how to determine polynomial equation vs. exponential equation

Answer 45

``` polynomial = finite differences exponential = ratio of 2nd differences ``` x | y | 1st dif. | 2nd dif. | RATIO* 0 | 1 | 2-1 = 1 | 7-1=6 | 6/6 = 1 1 | 2 | 9-2=7 | 13-7=6 | 6/6 = 1 2 | 9 | 22-9=13 | 19-13=6 | 3 | 22 | 41-22=19 | 4 | 41 | *aka constant ratio = base (a) of exponential function

Question 46

method of first principles to determine general derivative exponential formula

Answer 46

f(x) = 2^x k = lim (a^h -1)/h h->0 k = 2^0.0001 -1 / 0.0001 = 0.693 f'(x) = 0.693(2^x) k = f'(x) / f(x)

Question 47

inverse of an exponential function

Answer 47

logarithmic function | y = e^x y = log_e x

Question 48

natural logarithm

Answer 48

y'=lnx y=log_e x

Question 49

periodic vs. sinusoidal functions

Answer 49

repeats a pattern but can be in pieces, periodic; tanx, cscx, secx, cotx (ripples from a pepple in water) repeats a pattern and never ends periodic and sinusoidal; cosx, sinx (rope on end of pole up and down)

Question 50

chain rule for exponential functions

Answer 50

``` f(x) = a^(g(x)) -> f'(x) = a^x (lna)(g'(x)) f(x) = 3^(x^2-5x) -> f'(x) = 3^(x^2-5x)(ln3)(2x-5) ``` ``` y = e^x -> y' = e^x (lne)(y'(x)) -> y'(x) = e^x(y'(x)) y = e^(3x^2-8x) -> y' = (6x-8)e^(3x^2-8x) ```

Question 51

derivatives of sinx and cosx

Answer 51

``` f(x) = sinx -> f'(x) = cosx f(x) = cosx -> f'(x) = -sinx ```

Question 52

sin x vs. cos x

Answer 52

both have; local max=absolute max at y=1, local min=absolute min at y=-1, zeros (x-intercepts) occur at every π unit, one period every 2π units, domain {x element of all real numbers}, range {y element of all real numbers, -1= y, y= 1} sinx; y-intercept at (0,0) cosx; y-intercept at (0,1)

Question 53

rational functions

Answer 53

F(x) = f(x) / g(x) , g(x) ≠ 0

Question 54

chain rule for sinusoidal functions

Answer 54

f(x) = SIN(2x) 1. derivative of type, leave angle -> f'(x) = cos(2x) 2. multiply by derivative of angle -> f'(x) = (cos(2x))(2) 3. simplify -> f'(x) = 2cos(2x)

Question 55

second derivative

Answer 55

f(x) = 2x^3 - 6x^2 + x + 14 f'(x) = 6x^2 -12x + 1 f''(x) = 12x - 12 -determines inflection point -determines if other points beside inflection point are concave up or down -2 degrees less than original function i.e. f(x) = x^3 f''(x) = x^1

Question 56

inflection point

Answer 56

- where the slope is zero because the graph changes from positive or negative to the opposite (negative or positive) - enter 0 into second derivative to determine inflection point(s) - can have more than one

Question 57

original function's specification for first and second derivative

Answer 57

``` f(x) = increasing -> f'(x) = positive f(x) = decreasing -> f'(x) = negative ``` f(x) = 3rd degree (x^3) -> f'(x) = 2nd degree (x^2) -> f''(x) = 1st degree (x^1 aka x)

Question 58

equation of function

Answer 58

roots (aka x-intercepts) on a graph y-intercepts on a graph y-values of max/min/inflection points

Question 59

Method A

Answer 59

a

Question 60

Method B

Answer 60

b

Question 61

Even function vs odd function

Answer 61

c

Question 62

Standard form to determine equations

Answer 62

d

Question 63

rational function

Answer 63

- find asymptotes on graph - locate x & y intercepts - determine y values of max/min/inflection points - create table of values

Question 64

1st derivatives of rational functions

Answer 64

- locate max/min point | - intervals of increase/decrease

Question 65

2nd derivatives of rational functions

Answer 65

- locate any inflection points - find intervals of concavity - determine if critical number is max/min point