Calc Final (Made by Daniel)

Limit Exists

If left-hand limit = right-hand limit

Continuous Function Requirements

1. f(a) exists 2. limit exists 3. limit = f(a)

Difference Quotient

(f(x+h)-f(x))/h

Limit Definition of Derivative

f'(x)=lim(h→0)[(f(x+h)-f(x))/h]

How to Find f(x+h)

Replace every x with (x+h)

Conjugate Trick for Limits

Multiply by the opposite radical sign to remove square roots

Difference of Squares

a²-b²=(a-b)(a+b)

Direct Substitution Limits

If plugging in does not give 0/0, just substitute

0/0 Indeterminate Form

If plugging in gives 0/0, factor or simplify first

Domain Rule: Fractions

Denominator cannot equal 0

Domain Rule: Square Roots

Inside of even root must be ≥ 0

Domain Rule: Logs

Inside of logarithm must be > 0

How to Find Domain

Check denominators, radicals, and logs

If Differentiable Then Continuous

Differentiable ⇒ continuous

Power Rule for Derivatives

(x^n)'=nx^(n-1)

Derivative of Constant

0

Derivative of x

1

Derivative of e^x

(e^x)'=e^x

Derivative of ln(x)

(ln(x))'=1/x

Product Rule

(fg)'=f'g+fg'

Quotient Rule

(f/g)'=(f'g-fg')/g²

Chain Rule

Derivative of outside × derivative of inside

How to Recognize Product Rule

Two functions multiplied

How to Recognize Quotient Rule

One function divided by another

How to Recognize Chain Rule

Function inside another function

How to Rewrite Radicals

√x=x^(1/2)

How to Rewrite Denominators

1/x^n=x^(-n)

Equation of Tangent Line

y-y₁=m(x-x₁)

How to Find Tangent Line

1. Find derivative 2. Plug in x for slope 3. Find point 4. Use point-slope form

Velocity Definition

Velocity = derivative of position

Acceleration Definition

Acceleration = derivative of velocity

Average Velocity

(f(b)-f(a))/(b-a)

Position from Velocity

Integrate velocity function

Implicit Differentiation

Attach dy/dx whenever differentiating y

Derivative of y²

d/dx(y²)=2y(dy/dx)

Derivative of xy

x(dy/dx)+y

Critical Point Definition

Where f'(x)=0 or undefined

How to Find Critical Points

1. Find f'(x) 2. Set equal to 0 3. Solve 4. Check undefined points

Increasing vs Decreasing

f'(x)>0 increasing, f'(x)<0 decreasing

Concavity

f''(x)>0 concave up, f''(x)<0 concave down

Inflection Point

Where concavity changes

First Derivative Test

f' changes +→- = max, -→+ = min

Second Derivative Test

f''>0 minimum, f''<0 maximum

Optimization Steps

1. Write equation 2. Use one variable 3. Differentiate 4. Solve critical points 5. Test answers

Vertical Asymptote

Denominator = 0 and does not cancel

Horizontal Asymptote: Smaller Numerator Degree

If numerator degree < denominator degree, asymptote is y=0

Horizontal Asymptote: Equal Degrees

If degrees equal, use ratio of leading coefficients

Elasticity Formula

E(p)=-p(f'(p))/f(p)

Elastic Demand

E(p)>1

Inelastic Demand

E(p)<1

Unitary Demand

E(p)=1

Related Rates Steps

1. Differentiate with respect to time 2. Plug in values 3. Solve for unknown rate

Power Rule for Antiderivatives

∫x^n dx=(x^(n+1))/(n+1)+C

Special Antiderivative

∫(1/x)dx=ln|x|+C

Antiderivative of e^x

∫e^x dx=e^x+C

Indefinite Integral Meaning

Family of antiderivatives

Definite Integral Meaning

Signed area under the curve

How to Evaluate Definite Integrals

Antiderivative then upper bound minus lower bound

u-Substitution Recognition

Use when inside function and its derivative both appear

u-Substitution Steps

1. Let inside=u 2. Find du 3. Substitute 4. Integrate 5. Plug back in

How to Match du

If du=2x dx but problem has 8x dx, then 8x dx=4du

Riemann Sum Definition

Approximation of area using rectangles

Riemann Sum Width Formula

Δx=(b-a)/n

Right Endpoint Riemann Sum

Use right side of intervals for heights

Left Endpoint Riemann Sum

Use left side of intervals for heights

Area Under Curve

Definite integral gives signed area

Marginal Revenue

Derivative of revenue

Marginal Cost

Derivative of cost

Revenue from Marginal Revenue

Integrate marginal revenue

Profit Formula

Profit = Revenue - Cost

Integral of Marginal Function

Gives original total function

Exponential Growth/Decay Formula

A=A₀e^(kt)

Exponential Growth

k>0

Exponential Decay

k<0

Half-Life Definition

Time required for quantity to reduce by half

How to Solve Half-Life Problems

1. Plug known values into A=A₀e^(kt) 2. Solve for k 3. Plug new amount 4. Solve for time

Average Rate of Change

(f(b)-f(a))/(b-a)

How to Solve Factoring Limits

1. Factor 2. Cancel 3. Plug in

How to Recognize Removable Discontinuity

Factor cancels and leaves a hole

Motion Problems Strategy

Position → derivative gives velocity → derivative again gives acceleration

Definite Integral and Area

If graph is below x-axis, integral becomes negative

How to Check if Integral Equals Area

Function must stay above x-axis on interval

ln Rules

ln(ab)=ln(a)+ln(b)

ln Division Rule

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

Logarithmic Differentiation Steps

1. Take ln of both sides 2. Expand logs 3. Differentiate 4. Solve for y'

How to Solve Optimization Problems

Draw picture, write constraint equation, rewrite with one variable, differentiate

What Critical Points Can Be

Relative max, relative min, or neither

How to Find Relative Max/Min

Use first derivative test or second derivative test

When to Use Conjugates in Limits

When radicals create 0/0

How to Find Interval Width in Riemann Sums

(b-a)/n

Derivative of x^(-1)

-x^(-2)

Integral of x^(-1)

ln|x|+C

How to Recognize Definite Integrals

Integral has bounds

How to Recognize Indefinite Integrals

Integral has no bounds and answer needs +C