Math 111-001Final Exam Study Guide

May 5, 2015

1.1 - Four Ways to Represent a Function

• What is a function?

• What are the ways to describe a function?

1.2 - Mathematical Models: A Catalog of Essential Functions

• Polynomials, Power Functions, Rational Functions, Algebraic Functions, Trigonometric Functions, Exponential Func-tions, Logarithmic Functions.

1.3 - New Functions from Old Functions

• Transformations: Translation, Stretch/Shrink, Reflect

• Combinations: Sum, Difference, Product, Quotient

• Composition

1.5 - Exponential Functions

• f(x) = ax, a > 0

• Graphs

• Domain is (−∞,∞)

• e is number so that f ′(0) = 1

1.6 - Inverse Functions and Logarithms

• One-to-one and horizontal line test

• f−1(x) = y is the same as f(y) = x

• Cancellation equations

– f(f−1(x)) = x

– f−1(f(x)) = x

• Solve for inverses

• Logarithms and Laws of Logs

• Inverse Trig Functions

2.1 - The Tangent and Velocity Problems

• What is a tangent line?

• Velocity is Rate of Change of Position

2.2 - The Limit of a Function

• Find a limit with a table or a picture.

• When does the limit not exist?

• One-Sided Limits

• Infinite Limits

2.3 - Calculating Limits Using Limit Laws

• Limit Laws

• Bounding and The Squeeze Theorem

2.5 - Continuity

• 3 requirements:

– a in domain

– limx→a

f(x) exists

– limx→a

f(x) = f(a)

• Types of Discontinuities

• Intermediate Value Theorem

2.6 - Limits at Infinity: Horizontal Asymptotes

• Horizontal asymptotes

• limx→±∞

1

xr= 0

• Limits of other functions: trig, exponential, log, etc.

2.7 - Derivatives and Rates of Change

• Derivative at a point

f ′(a) = limx→a

f(x)− f(a)

x− a= lim

h→0

f(a+ h)− f(a)

h

2.8 - The Derivative of a Function

• Limit definition of the derivative

f ′(x) = limh→0

f(x+ h)− f(x)

h

• Differentiable/what is not differentiable?

• Differentiable implies continuous

• Higher derivatives

3.1 - Derivatives of Polynomials and Exponential Functions

• d

dxc = 0

• d

dxxn = nxn−1

• d

dxcf(x) = c

d

dxf(x)

• d

dx(f(x)± g(x)) = f ′(x)± g′(x)

• d

dxex = ex

3.2 - The Product and Quotient Rules

• d

dxf(x)g(x) = f ′(x)g(x) + f(x)g′(x)

• d

dx

f(x)

g(x)=f ′(x)g(x)− f(x)g′(x)

(g(x))2

3.3 - Derivatives of Trigonometric Functions

• Derivatives of sinx, cosx, tanx, secx, cscx, cotx

3.4 - The Chain Rule

• d

dx(f ◦ g)(x) =

d

dxf(g(x)) = f ′(g(x))g′(x)

• d

dxax = (ln a)ax

3.5 - Implicit Differentiation

• How to find slopes of tangents when the variables are mixed together.

3.6 - Derivatives of Logarithmic Functions

• d

dxloga x =

1

x ln a

• d

dxlnx =

1

x

• Logarithmic Differentiation

3.7 - Rates of Change in the Natural and Social Sciences

• How derivatives can help in science.

• Marginal cost, demand, revenue, etc.

3.8 - Exponential Growth and Decay

• Proportional growth/decay

• Compound interest

4.1 - Maximum and Minimum Values

• Extreme values

• Local max/min

• The Extreme Value Theorem

• Fermat’s Theorem and Critical points

4.2 - The Mean Value Theorem

• Rolle’s Theorem

• Mean Value Theorem

4.3 - How Derivatives Affect the Shape of a Graph

• Increasing/Decreasing Test

• First Derivative Test

• Concavity Test

• Second Derivative Test

4.4 - Indeterminate Forms and l’Hospital’s Rule

• l’Hospitals Rule: Use when in form 00 or ∞∞

• Indeterminate forms

4.5 - Summary of Curve Sketching

• Use everything you can to sketch.

4.7 - Optimization Problems

• Draw pictures

• Write equations that represent known information

• Use equations to write an equation for the value you want to maximize/minimize.

• Use the derivative to maximize/minimize.

• Make sure you answered the question!

4.9 - Antiderivatives

• Antiderivatives are the opposite of the derivative

5.1 - Areas and Distances

• Use a number of boxes to approximate area.

• ∆x = b−an , xi = a+ i∆x

• Rn = ∆xf(x1) + ∆xf(x2) + . . .+ ∆xf(xn)

• Ln = ∆xf(x0) + ∆xf(x1) + . . .+ ∆xf(xn−1)

5.2 - The Definite Integral

• Riemann sum computations for definite integrals

• Properties of definite integrals

5.3 - The Fundamental Theorem of Calculus

• The Fundamental Theorem of Calculus parts 1 and 2

5.4 - Indefinite Integrals and the Net Change Theorem

• Definite integrals - notation for antiderivatives

• Net Change Theorem

5.5 - The Substitution Rule

• Undo the chain rule

• Look for an “Inside Function”

6.1 - Areas Between Curves

• Find where they touch - set functions equal to one another

• Bigger - Smaller

6.2 - Volumes

• Rotate f(x) about x-axis for a ≤ x ≤ b

V =

∫ b

aπ(f(x))2 dx

• Rotate area below f(x) above g(x) about x-axis for a ≤ x ≤ b

V =

∫ b

aπ((f(x))2 − (g(x))2) dx