AP CALC AB UNIT 1

what line is average rate of change asking to find the slope of ?

what line is average rate of change asking to find the slope of ?

the slope of the secant line which is the direct line going thru two points.

what is the formula for average rate of change?

it is just normal slope. f(a) - f(b) / a - b

what line do you find the slope for the instantaneous rate of change?

find slope of tangent line. it will find get the exact rate of change at 1 point

how to find instantaneous rate of change or exact rate of change?

take the derivative or limit

what is the formula for the limit of exact rate of change

lim h → 0 f(a+h) - f(a) / h

what does this limit mean. lim x → 2 f(x) = 4

as I come closer to x = 2, the y value comes closer to 4 on both sides.

what does the negative (-) on top of the number in a limit represent?

it is a left hand side limit

what does the positive (+) on top of the number in a limit represent?

it is a right hand side limit

what needs to be true in order for the general limit to exist?

LHL exists. RHL exists. and LHL = RHL

the basics of limit rules.

1. a limit of a constant is a constant 2. you can split up sums, differences, products and quotients into separate limits and then solve. 3. same with powers, you can take the power after solving the more simple part

methods for solving a limit

1. direct sub (only if no domain restriction. cannot have a 0 in denominator)

2. factor

3. conjugate method

4. trig limits

5. lhopitals rule

what is jump discontinuity?

It occurs when the left-hand and right-hand limits exist but are not equal at the point of discontinuity. There is a gap between both points.

what is infinite discontinuity?

The limit of a function approaches infinity or negative infinity. Seen with functions with vertical asymptotes.

what is point/removable discontinuity?

this is where there is a hole in the function and is found by finding a hole in the equation.

what does continuity mean in calculus.

graph can be drawn without lifting the pen from the paper.

how to prove/check continuity. f is continuous at x = c if

f(c) exists, lim approaching c of f(x) exists, the limit approaching c of f(x) = f(c)

horizontal asymptotes, when lim approaches infinity: if largest exp in N < largest exp in D then what happens

lim = 0

horizontal asymptotes, when lim approaches infinity: if largest exp in N > largest exp in D then what happens

lim = DNE

horizontal asymptotes, when lim approaches infinity: if largest exp in N = largest exp in D then what happens

ratio of leading coefficients

what is the intermediate value theorem?

if the function is continuous on [a,b] and y is between f(a) and f(b) then there exists at least one number (x-value) x = c in the open interval (a,b) such that f(c) = y