AP Calc Unit 2

f'(3)= (-2pi - 3) / 8

Chart says g(3) = 4 g'(3) = -2 h(3) = 3 h'(3) = pi f(x)= { g(x) - h(x) } / g(x) find f'(3)

CONTINUITY

DIFFERENTIABLE =

- csc^2 x

Derivative of cot x

product rule

Derivative of the first function times the second function plus the first function times the derivative of the second function. f ' [ f(x)g(x) ] = f(x)g'(x) + g(x)f'(x)

Exponent Trig Angle

acronym for trig derivatives

slope

average rate of change

yes. in that case you just take the inner and apply the chain rule onto that

can there be more than one chain rule in a question? multiple? explain

chain rule

d/dx [ f(g(x)) ] = f'(g(x)) * g'(x)

- sin x

derivative of cos x

-csc x cot x

derivative of csc x

sec x tan x

derivative of sec x

cos x

derivative of sin x

sec^2 x

derivative of tan x

f(x+h) = y2 f(x) = y1 h = x2 - x1

describe the components in the derivative formula

f'(x), dy/dx, d/dx f(x), tangent line

different ways to say derivative

no

do u use the product rule for coefficients

at sharp corner (like absolute value), at cusp, at a vertical tangent line (like the image at x=0), at a discontinuity (like a floor or cieling function)

examples of where function isn't differentiable

first you find the derivative which is the slope of the function everywhere. next, the question asks for the point x=3 so you plug in 3 into derivative to find the slope at 3. so our equation looks like y = 3x + b. Then you plug in 3 into the original equation to find a coordinate point (3 , 43) which you plug in to find b. and your equation is y=3x+34. (or you could use point slope form)

explain how to do this. find the equation of the line tangent to the curve y=2x^3-2x^2+4x-5 at x=3

f ' (x) = 6x

f(x) = 3x^2, find derivative

f ' (2) = 3 (you first find the derivative and then plug in 2 for x in the equation for the derivative.)

f(x) = x^2 - x , find f ' (2) HOW DO U DO IT

(pi/4 , 1) and (-pi/4, -1)

find all the points on the curve f(x)=tanx where the tangent line is parallel to the line y=2x on the interval (-pi/2 , pi/2)

(- x^2 cos x) - (2x sin x) don't forget the negative in the front

find the derivative of 4 - (x^2 sin x)

-2x sin x

find the derivative of [ (2x cos x) - (2 sin x) ]

(first realize it is a power rule. and once you are doing the power rule you need to use the chain rule.) 2sec^2(2x)sin(3x) + 3cos(3x)tan(2x)

find the derivative of y = sin(3x)tan(2x)

so basically it says find coordinates where derivative is = to 0 because if the line is horizontal the derivative is 0. so you find the derivate and set it equal to zero. then you find the zeros of the derivative function and you plug those x-values back into the orignal function to get your y-values and you have your coordinate points which are, (0,4) and (+- sqrt3 , -5)

find the points on the graph of f(x) = x^4 - 6x^2 + 4 where the tangent line is horizontal

8 (Basically this question is asking to find the slope of the derivative of x^2+2x at x=3.)

find the slope of the tangent line of f(x) = x^2 + 2x when x = 3

(0,1) (1,0) (-1,0)

for the function give all the points for which f'(x)=0 and where f'(x) DNE f(x) = cube root of [ (x^2 -1) ]^2

plug in H into all the piecewise equations and if the value is the same then the function is continuous.

how to find out if piecewise function is continuous at x = H? (H being a constant)

first it has to be continuous, so check that. then the derivate of all the equations have to be the same.

how to find out if piecewise function is differentiable at x=H? (H being a constant)

you find the derivative and then you find the derivative of the derivative

how to find the second derivative

f'(x) = 3 (you could use derivative formula or you could use power rule.)

if f(x) = 3x - 2 , find f'(x) AND TELL ME HOW YOU FOUND IT

power rule

if h is a rational number and the defined function x^n is differentiable then derivative of x^n = n(x)^(n-1)

you can recognize, in the derivative formula, what the equation of the derivative is and then you can find an easier way to solve that derivative.

if i'm given the long long formula with derivative in it then what can i do?

zero

if the function is a constant the derivative is always _______

no

if the function is continuous does that mean it is also differentiable

yes

if the function is differentiable does that mean it is also continuous

rewrite. it is easier to do product rule than quotient rule

if you have fraction that you can rewrite so you can use the product rule than should you rewrite it and use product or quotient rule?

yes

if, in the long long formula of derivative, you replace h with y or r is it the same thing?

no

is the slope constant in a quadratic function

derivative formula

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

tangent line

normal line is perpendicular to ___________ ______

f''(x)

second derivative notation

sin (x^5) just x is to the fifth

sin x^5 equals

(sin x)^5 quantity to the fifth. even the sin.

sin^5 (x) equals

0

slope = 0 of this line so the derivative of this line is...?

rewrite

sometimes you have to ______ a function so it looks like x^n so you can use the power rule

do the derivative of the outermost function first with the inner function inside, left alone. then multiply that by the derivative of the inner function.

summary of chain rule

secant line

the line through two points on a curve

secant on a curve

the line through two points on a curve

constant ; derivative ; derivative

the slope of a linear function is __________ so it is equal to the derivative. So y = mx + b, m is the slope and the _________

if the limit as x approaches c is equal to f(c) and if the limits from the right and left side of derivate are equal

two rules for function to be differentiable are...

you do the same steps as the tagent line but when you get the slope at the given x point you find its negative recipricol since a normal line is perpendicular to the tangent line.

what about finding the equation of a normal line

0

what is the derivative of y = xpi^2, x being a constant

0

what is the derivative of y=7

0

what is the slope of this line

yes

would you use product rule for finding the derivative of x^2 * sinx

quotient rule for derivatives

{ g(x)f'(x)-f(x)g'(x) } / g(x)^2 loq Dhigh - high Dlow over low low