MATH 112 FINAL BYU

derivative of a^x

(a^x)lna

Definition of integrals

(int)a-b f(x) dx = lim(n->inf) nE(i=1) f(xi) (delta)x delta x = b-a/n right endpoint xi = a +(delta x)i

Def of Derivative: What are the two limit definitions

(lim h→0) [f(a +h) - a]/ h OR (lim x→a) [f(x)-f(a)]/[x-a]

limx→0 [cosx - 1]/[x] = ?

0

limx→0 sinx/ x

1

Implicit Differentiation: xy = cosy

1. 1*y + x*(dy/dx) = -siny*dy/dx 2. (dy/dx). [x + siny] = -y 3. dy/dx = -y/(x+siny)

To find the absolute max or min of f over and interval

1. Find the critical numbers of f in (a,b). 2. Find the values of f at the critical numbers in (a,b). 3. Find the values of f at the endpoints. 4. Determine the largest and smallest values of f from the previous 2 steps.

Limits: Horizontal Limits

1. If the highest exponentials are the same then lim is the ratio of the leading coefficients. 2. If Numerator is higher then it approaches infinity (+or-) 3. if denominator is higher it approaches 0

Related Rates Problems:

1. Read the problem carefully. 2. Draw a diagram even if it is not requested for the problem. 3. Introduce notation and assign symbols to all quantities that are functions of time. 4. Express the given information and required rates in terms of derivatives. 5. Write an equation that relates the various quantities. Use geometry to reduce one of the variables if needed. 6. Use the chain rule to differentiate with respect to t. 7. Substitute given information into the equation and solve for the unknown rate.

Optimization

1. Read the problem carefully. 2. Draw a diagram if possible and label the quantities. 3. Introduce Notation. 4. Express the quantities as functions and/or equations. 5. If there are two variables use the constraint to reduce the problem to one variable. 6. Now the function should have one variable so find the global maximum and/or minimum for the problem. 7. Make sure you answer the problem that was asked! For example, if the problem asks for the perimeter don't give the dimensions as your answer.

what do you do to take the derivative of y = x^x?

1. Take the natural logarithm of both sides of an equation and use the laws of logarithms to simplify the expression. 2. Differentiate both sides implicitly with respect to x. 3. Solve for y' [You don't need to reverse the log at the end]

Rolles Theorem

1. f is continuous on the interval [a,b]. 2. f is differentiable on the interval (a,b). 3. f(a) = f(b). Then there is a number c in (a,b) such that f0(c) = 0.

d/dx (loga x) =

1/ x lna

Derrivative of arctan(x)

1/(x^2 +1)

derivative of lnx

1/x

Derivative of arcsec(x)

1/x√{x^2-1}

Derivative of arcsin(x)

1/√{1-x^2}

sin(2x)

2sin(x)cos(x)

Limits: Find lim x→4-- (4−x)/ |4−x| .

DOES NOT EXIST

Limits:Find limit of Piece wise functions.

For limit to exit then both sided limits have to be equal.

Squeeze Theorem

IF f(x) <= g(x) <= h(x) AND limf(x) = limh(x) = L THEN limg(x) = L

L'Hospitals rule

IF: lim fx/gx = [Inf/Inf or 0/0 or 0*inf or 0^0 or inf^0 or 1^inf] THEN: == lim f'x/g'x

Derivative

Lim(h->0) [f(a+h) - f(a)]/h

Calculating Derivatives: Find derivative of (x^2 +1)^(1/x)

Logarithmic differentiation: 1. lny = (1/x)*ln(x^2 +1) 2. 1/y *y' = (-1/x^2)(ln(x^2 +1) + (1/x)*(1/(x^2 +1)*2x) 3. y' = y*[(-1/x^2)(ln(x^2 +1) + (1/x)*(1/(x^2 +1)*2x)]

Calculating Derivatives: Basic Rules

Product Rule, f'(x)g(x) + g'(x)f(x) Quotient Rule: Lo dhi - hi di / lo lo Power Rule: x^a ==> a*x^(a-1)

Intermediate Value Theorem

Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b) where f(a) != f(b). Then there exists a number c in (a,b) such that f(c) = N.

Calculating Derivatives: Find derivative of [3e^x +2x]/sinx

[sinx*(3e^x +2) - (3e^x +2x)*(cosx)]/ sin^2(x)

Calculating Derivatives: Find d/dx{arccos(2x)}

arccos ==> -1/[√(1-x^2)]

cos(A+B)

cosAcosB-sinAsinB

cos(2x)

cos^2(x) - sin^2(x)

distance formula

d^2 = (x - x1) + (y - y1)

Calculating Derivatives: Find derivative of f(x) = (x^2)(cos√((e^2x)+1)

do power rule and chain rule: (2x)*((cos√((e^2x)+1) + (x^2)*(-sin√((e^2x)+1) *1/2(e^2x)+1)*(2e^2x)

Mean Value Theorem:

f'(c) = f(b)−f(a)/ b−a

Critical numbers

f'(x) = 0 or the end points points or f'x = dne

first derivative test

if c is a critical number and f'(c) = 0, then f has a local maximum at c if f' changes from positive to negative, and f has a local minimum at c if f' changes from negative to positive.

second derivative test

if f'(c) = 0 and f''(c) > 0, then f has a local minimum at c, and if f'(c) = 0 and f''(c) < 0, then f has a local maximum at c. Do not mix these two up!

Integral[a to b] of f(x)dx =

lim n→∞ n {sum} i=1 (f(x∗ i)δx.)

limit definition of Derivative

lim x→a f(x)−f(a) / x−a

Limits: a) Find (lim x→2−) x /x−2 .

negative infinity

Limits: lim-x→π /2+ tanx /x

negative infinity

sin(A + B)

sinAcosB + cosAsinB

Trig Identities

sin^2 x + cos^2 x = 1 tan^2 x + 1 = sec^2 x 1 + cot^2 x = csc^2 x

Calculating Derivatives: Find f'(x) if f(x) = 7^x^2

use the exponent rule and chain rule. 7^x^2 = (7^x)*ln7*2x

Newtons Method

x(n+1) = xn - [f(xn)]/[f'(xn)]

Anti Derivative of 3x^2 + 2/x^2

x^3 -2/x

point slop form

y= m(x-x1) +y1

cos(x/2)

±√((1+cosx)/2)

sin(x/2)

±√((1-cosx)/2)