L’Hopital’s Rule:

In plain English: if a limit results in an indeterminant form, the limit is equal to the derivative of both the numerator and denominator.

If f(a)/g(a) = 0/0 or =∞/∞, then: lim x→a f(x)/g(x) = lim x→a f’(x)/g’(x)

Average Rate of Change (slope of the secant line)

In plain English: The average rate of change is similar to the basic Δy/Δx*, but instead of y values in the numerator, there are slope values, and x values are on the bottom.*

If the points (a, f(a)) and (b, f(b)) are on the graph of f(x) the average rate of change of f(x) on the interval [a,b] is: [f(b) - f(a)]/[b-a]

Limit Definition of the Derivative (slope of the tangent line)

In plain English: If you take two points, and move them so close together you are almost (but not quite) adding 0, you create a tangent line and have therefore found the instantaneous rate of change.

f’(x) = lim h→0 [f(x+h) - f(x)]/h

Common derivatives YOU MUST KNOW !!!!!!!!!!

Properties of Log and Ln:

ln 1 = 0

ln e^a = a

e^ln x = x

ln x^n = n ln x

ln(ab) = ln a + ln b

ln(a/b) = ln a - ln b

Differentiation rules:

CHAIN RULE: d/dx [f(u)] = f’(u) du / dx

PRODUCT RULE: d/dx (uv) = (u)(dv/dx) + (v)(du/dx)

QUOTIENT RULE: d/dx(u/v) = [(v)(du/dx) - (u)(dv/dx)'] / v^2

Mean Value Theorem (MVT):

In plain English: If f(x) is made up of one going line, and there exists two y values in two coordinates (say 3 and 5), there has to be an x value where the y value is in between 3 and 5 (like 4) because the lie does not “jump” or have holes.

If the function f(x) is continuous on [a,b] and the first derivative exists on the interval (a,b) then there exists a number x=c on (a,b) such that f’(c) = [f(b)-f(a)]/[b-a].

Curve Sketching and Analysis:

To find a local minimum: f’(x) changes from - to +.

To find a local maximum: f’(x) changes from + to -.

To find an absolute max/min: compile all the local points and plug back into the original function.

To find a point of inflection: f’’(x) goes from + to - or - to +.

• • f’’(x) is concave up

• • f’’(x) is concave down

The Fundamental Theorem of Calculus:

a to b∫f(x)dx = F(b) - F(a) where F’(x) = f(x)

The 2nd Fundamental Theorem of Calculus:

In plain English: The derivative of an integral where one of the end points is a function (A) is that function plugged directly into the function in the integral (B) multiplied by the derivative of the end point function (A’)

d/dx (# to g(x)) ∫f(x)dx = f(g(x)) * g’(x)

Average Value:

**If the function f(x) is continuous on [a.b] and the first derivative exists on the interval (a,b), then there exists a number x=c on (a,b) such that the average value is f(c) = 1/[b-a] ***∫f(x)dx ***note the integral is from a to b

Euler’s Method: Create the table as shown. Fill out all information to find your value. Note that Δx is the “step” you are talking, and dy/dx is the slope.

Logistics Curves:

P(t) = L / [1+ Ce^(-Lk)t] where L is carrying capacity.

• maximum growth rate occures when P = 1/2 L

dP/dt = kP(L - P) = (Lk)(P)(1-[P/L])

Integrals YOU MUST KNOW !!!!!!!!!!!!!!!

Integration by Parts:

∫u dv = uv - ∫v du

USE LIPET TO DETERMINE U:

• Logs

• Inverse trig

• Polynomial functions

• Exponential functions

• Trig

Arc Length:

For a function f(x): L = ∫[(1+f’(x)^2]^1/2 dx

For a polar graph r(θ): L = [(r(θ)^2+(r’(θ)^2]^1/2 dθ

Lagrange Error Bound:

Distance, velocity, acceleration

Velocity = d/dt (position)

Acceleration = d/dt (velocity)

Velocity vector = <dx/dt, dy/td>

Speed = |v(t)| = [(x’)^2+(y’)^2]^1/2 <-- this is Pythagorean theorem !!!!!

Distance traveled = ∫speed

Polar curves:

For a polar curve r(θ), the area in one part is 0.5∫r(θ)^2 dθ.

dy/dx = dy/dθ * dθ/dx = d/dθ [r(θ)sinθ] / d/dθ [r(θ)cosθ]

Remember that x = rcosθ and y=rsinθ

Ratio Test:

The series of a(n) converges if lim n→infinity |a(n+1)/a(n)| < 1. CHECK ENDPOINTS!

Alternating Series Error Bound:

The maximum error of a convergent alternating series is the next term of the sequence.

P-Series:

Volume

Disk method: V = π∫[R(x)]^2 dx

Washer method: V = π∫[R(x)^2 - r(x)^2] dx

Known cross sections

V = ∫A(x) dx or ∫A(y) dy

Taylor series:

f(x) is about f(c) + f’c(x-c) + f’’(c) 8 (x-c)^2 / 2! _