AP Calculus AB

Find derivative at a point to find the slope. Plug in the point to the ORIGINAL equation to find the point of tangency (your x and y values) then use point slope form (y-y1=m(x-x1)

How do you find an equation of a tangent line when given an x value?

absolute value of velocity

How do you find speed?

set velocity equal to zero or set the derivative of position equal to zero (v(t)=0 or s'(t)=0) (BE AWARE THAT when explaining why a point is max or min, you must explain the sign change. Not that it equals zero!)

How do you find the maximum or minimum height of a function? (2 ways)

Make a sign chart to determine behavior; plot all zeros

How do you graph f if you're given a graph of f' or vice versa?

must be continuous at the point and the left and right hand derivatives must be equal

How do you know if a function is differentiable at a point? (2 conditions)

vertical line test

How do you see if something is a function?

left

If velocity is negative, then P (particle) is moving...

divide b by 2 and then square. Add to both sides.

how do you complete the square?

switch x and y solve for y

how do you find the inverse of a function?

squeee theorem

for sin and cosine limit functions, you can use the __ if f(x) is less than or equal to g(x) and is less than or equal to h(x) for all x on [a,b] and lim x->c h(x)=limx->c f(x)=L then lim x->c g(x)=L

lim x-> infinite f(x) = a

the line y=a is a horizontal asymptote of f(x) if

0

to find the turning points of a graph, set the derivative equal to

f'(a)*change in x

using linearization, how do you find change in f using L(x)?

logarithmic, inverse, polynomial, exponential, trigonometric

what are the 5 basic classes of functions?

a horizontal line

what is the derivative graph of a linear function?

y=3√x (cube root of x)

what kind of function is this and what is the equation

y=cos x

what kind of function is this and what is the equation

y=e^x

what kind of function is this and what is the equation

y=x^2 parabolic

what kind of function is this and what is the equation

y=x^3 cubic

what kind of function is this and what is the equation

y=|x|

what kind of function is this and what is the equation

y=√x

what kind of function is this and what is the equation

y=1/x

what kind of function is this and what is the equation (1/x^2 would be similar to this, but all would be above x axis)

positive

when a function increases, the derivative is...

jump discontinuity

you have a __ if right and left hand limits do not equal

second

Does the second or first derivative determine concavity?

f'(a)=lim as x->a of f(x)-f(a)/x-a

How do you find a derivative at a point? (Definition of derivative at a point)

Find the derivative

How do you find a slope of a tangent line?

v(b)-v(a)/b-a (v is velocity)

How to find average acceleration?

a ( The given limit is the derivativ of g(x) at x=3. Since g'(x) is negative, this means the behavior of the graph is decreasing.

If lim x->3 g(3)-g(x)/3-x = -0.628, then near the point where x=3, the graph of g(x) a. is decreasing b. is increasing c. is concave downwards d. has a point of inflection

Increasing concave up (velocity being positive indicates increasing, acceleration being positive indicates concave up)

If the position of a particle is denoted by s(t), as a funcgion of yime, if the particle's velocity and acceleration are both positive, what is the begavior and concavity of the fuction?

right

If velocity is positive, then P (particle) is moving...

even

If you put in the opposite of a number into a function and get the same number out, what kind of function is it?

c

Lim h->0 tan(x(x+h))-tan(2x)/h is a 0 b sec^2 (2x) c 2sec^2(2x) d nonexistent

s'(t) derivative of position function

The instantaneous velocity function is lim b->a s(b)-s(a)/b-a. What is another way to find instantaneous velocity?

true

True or false: an nth degree polynomial will have n+1 terms and at MOST n-1 turning points

v'(t) or s"(t)

What are two ways to find acceleration?

v(t)=lim b->a s(b)-s(a)/b-a

What is the instantaneous velocity function?

s(t)=s1+v1t+0.5a1t^2 (s1= initial position, v1=initial velocity a1= initial acceleration t=time)

What is the position function?

Y/x

What is the rario for a tangent value?

X/r

What is the ratio for a cosine value?

Y/r

What is the ratio for a sine value?

lim as h-->0 of f(x+h)-f(x)/h (you plug in x+h anywhere in the equation that has an x)

What's the definition of a derivative?

negative

When a function decreases, the derivative is....

When velocity and acceleration are different signs

When is speed decreasing?

When both velocity and acceleration are positive

When is speed increasing?

Mean value theroem

When the average rate of change equals the instantaneous rate of change (and is continuous and differentiable) use __ to find how many times they equal each other

v(t)=v1-a1t (v1 is initial velocity. a1 is initial acceleration, which if you're talking about earth's gravity, it will be 9.8m/s^2 or 32ft/s^2

You can find instantaneous velocty by using the function lim b->a s(b)-s(a)/b-a or finding the derivative of the position function. HOWEVER, when finding the velocity of an object rising or falling under the influence of GRAVITY near the earth's surface (ignoring air resistance) you must use the function __

1

cos^2∅+sin^2∅=?

-sinx

derivative of cos x

1/|u|√u^2-1 * du/dx

derivative of inverse sec

cosx

derivative of sin x

continuous

f(x) is continuous @ x=a if lim x-> a+ f(x)=lim x->a- f(x)= f(a)

c (because the definition of a derivative says that if a function x->c = a then f'(c)=a )

given function g such that lim h->0 g(x+h)-g(x-h)/h = 6-4x which of the following statements would be true? I g'(0)=6 II g"(0) <0 III g"'(0)=0 a. I and II only b. I and III only c. II and III only d. I, II, and III

s=theta/360*2pir if in degrees s=theta*r if in radians

how do you find the length of a part of a circle?

f'(a)=0

if f is a function such that lim x->a f(x)-f(a)/x-a = 0 which of the following must be true? a. lim x->a does not exist b. f'(a)=0 c. f(a)=0 d. f(x) is continuous at x=0

every y value between m and n must occur at least once, at some x value between a and b

if f(x), a function, is continuous on [a,b] and f(a)=m and f(b)=n then....__ according to the intermediate value theorem

an * x^n-1

if f(x)= a * x^n,then f'(x)=

no asymptote

if the degree of the denominator is greater than the degree of the denominator, the horizontal asymptote is at

a/b (a is coef of numerator/coef of denom)

if the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is at

y=0

if the degree of the numerator is smaller than the degree of the denominator, then the horizontal asymptote is at

discontinuity

if there is a vertical asymptote, there is a __

odd

if you put in the opposite number and get out a different result, then you have what kind of function?

a^2=b^2+c^2-2bc*cosA (A is an angle)

law of cosines

sinA/a=sinB/b=sinC/c

law of sines

0

lim h->0 cos h-1/h=?

1

lim h->0 sinh/h =?

1

lim x->0 sinx/x =?

logaB-logaC

loga(B/C) = ? properties of log

logaB+logaC

loga(BC) = ? properties of log

u*dv+v*du

product rule d/dx (u*v)

v*du-u*dv/v^2

quotient rule d/dx (u/v)

c limx->a f(x)

the constant multiple law says that the lim x-> a of c f(x) =?

f'(c)

the mean value theorem states that if f(x) is differentiable on (a,b), then there exists at least one value c in (a,b) such that f(b)-f(a)/b-a=

true

true or false: sum law says that when you're finding the limit of a function subtracting, adding, multiplying, or dividing another limit, you take the limit of each one individually ( ex: lim x->2 5x^2 - 7x turns to lim x->2 5x^2 - limx->2 (7x)

false ( a function is continuous if it is differentiable at x=c, but continuity does not mean differentiability)

true or false: If a function is differentiable at x=c then it is continuous at x=c. Similarily, if a function is continuous at x=c, then it is differentiable at x=c.

true

true or false: a derivative of a constant is always zero

false

true or false: continuous functions cannot have discontinuity outside of the domain

true

true or false: exponential functions have similar shapes to their derivatives

true

true or false: polynomials are fully continuous for all real numbers

true

true or false: the biggest cosine can ever be is 1

false (think of a car going in one direction. Just because it stops doesn't mean it's turning around)

true or false: zero velocity DEFINITELY means that a reversal in direction has occured

y=ln x

what kind of function is this and what is the equation

y=sinx

what kind of function is this and what is the equation

y=tanx (vertical asymptotes occur at pi/2)

what kind of function is this and what is the equation

y=x linear

what kind of function is this and what is the equation

f'(g(x))*g'(x)

d/dx f(g(x)) by the chain rule is equal to

sec^2theta

tan^2∅+1=?

a^u * lna * du/dx

derivative of a^u (a being any constant)

-csc^2x

derivative of cotx

-cscxcotx

derivative of csc x

e^u * du/dx

derivative of e^u

-1/√1-(u)^2 * du/dx

derivative of inverse cos u

-1/1+u^2 * du/dx

derivative of inverse cot

-1/|u|√u^2-1 * du/dx

derivative of inverse csc u

1/√1-(u)^2 * du/dx

derivative of inverse sin u

1/1+(u)^2 * du/dx

derivative of inverse tan u

1/u * du/dx

derivative of ln u

1/u * 1/lna * du/dx

derivative of log a u

secxtanx

derivative of secx

sec^2x

derivative of tanx

root (x2-x1)^2 + (y2-y1)^2

distance formula

(x-h)^2+(y-k)^2 = r^2 ( h and k are the origin points)

equation of a circle

csc^2theta

1+cot^2∅