Precalculus Review.
Please print the prelecture notes before the lecture: Review PreLecture (Review PostLecture)Fundamental to a student's success in a mathematics course is their familiarity with the prerequisite material. The topics and problems listed below represent many, but not all, of the perquisites necessary to take on the material in MATH 125.
- Suggested Problems are:
- Lecture Videos:
| Inequalities | Section 1.1 | # 13-31 |
| Function Basics | Section 1.1 | # 39-59, 64-77 |
| Function Symmetry | Section 1.1 | # 60-63, 78-81 |
| Linear Functions | Section 1.2 | # 1-31 |
| Quadratic Functions | Section 1.2 | # 35-47 |
| Function Domain | Section 1.3 | # 1-12, 40 |
| Function Composition | Section 1.3 | # 27-34 |
| Piecewise Functions | Section 1.3 | # 35-38 |
| Trigonometric Functions | Section 1.4 | # 1-17 |
| Inverting 1-to-1 Functions | Section 1.4 | # 1-17 |
| Inverting Trigonometric Functions | Section 1.4 | # 33-54 |
| Exponential Functions | Review Exercises | # 1-10, 29-32, 42 |
| Logarithmic Functions | Review Exercises | # 11-28, 33, 34 |
Introduction To Calculus I
Precalculus Review, Functions
Laws of Expoenent and Logarithms
Trigonometric Functions
Piecewise-defined
Modeling with Mathematics
Section 2.1: The Limit Idea: Instantaneous Velocity and Tangent Lines.
Please print the prelecture notes before the lecture: Section 2.1 PreLecture (Section 2.1 PostLecture)Learning objectives are
- Understand the connection between rates of change, tangent and secant lines, and velocity.
- Understand how tangent lines are connected to secant lines.
- Learn the terms average and instantaneous and understand the mathematical expressions associated with them.
- Suggested Practice in this section:
- Lecture Videos:
|
Averages/Secant Lines |
# 1, 7-10, 22, 25, 26 |
|
Estimating Instantaneous Rates of Change |
# 22, 24-26 |
Tangent Lines and Secant
Velocity
Section 2.2: Investigating Limits.
Please print the prelecture notes before the lecture: Section 2.2 PreLecture (Section 2.2 PostLecture)Learning objectives are
- Given the graph of a function, be able to estimate a one or two-sided limit, if it exists.
- Understand that the actual value, f(a), and the limit as x approaches a are unrelated; they don't necessarily exist or equal one another.
- Understand that infinity is
NOT a number and is used as a description of unbounded growth. - Suggested Practice in this section:
- Lecture Videos:
|
Estimating Limit Values |
# 1-4 |
|
Limits from Graphs |
# 5, 6, 40, 49-56 |
Limit
Numerical Approach
Section 2.3 Basic Limit Laws.
Please print the prelecture notes before the lecture: Section 2.3 PreLecture (Section 2.3 PostLecture)Learning objectives are
- Identify functions and values for which the basic limit laws apply.
- Use the five basic limit laws to evaluate limits.
- Suggested Practice in this section:
- Lecture Videos:
- Understand the definition and develop a graphical understanding of continuity.
- Learn how continuity is preserved through operations and learn the discontinuities of common functions.
- Identify and classify the discontinuities of a function presented graphically or algebraically.
- Suggested Practice in this section:
- Lecture Videos:
- Understand the definition of form and identify the seven indeterminate forms.
- Learn and master the first three techniques for calculating limits: Direct Substitution, Simplification, and Conjugation.
- Learn how to calculate the limit of a piecewise function. Recall that absolute value function are piecewise functions.
- Suggested Practice in this section:
- Lecture Videos:
- This is covered in Chapter 3.
- Learn and apply the Squeeze Theorem to limits of functions which are easy to bound.
- Recognize limit problems which can be solved by utilizing the new limit identities sin(x)/x and (cos(x)-1)/x as x approaches 0.
- Extra Videos for Practice (Main lecture videos are in Section 3.6)
- Learn the term asymptote and master identifying both vertical and horizontal asymptotes graphically and algebraically.
- Learn the asymptotes of common functions.
- Suggested Practice in this section:
- Lecture Videos:
- Understand and apply the Intermediate Value Theorem; in particular, understand that a continuous function on a closed domain attains every intermediate value.
- Use the Intermediate Value Theorem to isolate roots of common functions.
- Suggested Practice in this section:
- Enjoy this short Numberphile video on an interesting application of the Intermediate Value Theorem.
- Lecture Videos:
- Understand how a tangent line results from the limit of secant lines.
- Learn how average rates of change, the slope of secant lines, and average velocity are connected.
- Learn how instantaneous rates of change, the slope of tangent lines, and instantaneous velocity are connected.
- Learn the two equivalent definitions of the derivative.
- Lecture Videos:
- Learn the definition of the derivative function.
- Master the usage of the Lagrange and Leibniz notation for derivatives.
- Learn how a function can fail to be non-differentiable; in particular, identify the points which are non-differentiable on a graph.
- Recognize the graph of a function f and its derivative f '.
- Understand that differentiability implies continuity and that continuity does not imply differentiability.
- Master the Power, Constant Multiple, Sum/Difference Rules for Differentiation.
- Lecture Videos:
- Master the Product and Quotient Rules of Differentiation; in particular, learn how to combine these rules with those developed in section 3.2.
- Lecture Videos:
- Identify and describe derivatives and rates of change in examples from the natural and social sciences. In particular, become familiar with the concept of marginal cost and Galileo's formula for gravitational movement.
- Use specific examples from the sciences to demonstrate understanding of terms from differential calculus.
- Lecture Videos:
- Understand that higher derivatives are defined by successive differentiation.
- For nicely behaved functions, derive a formula for the n-th derivative.
- Understand the role the second derivative plays on the graph of a function; in particular, the second derivative measures how fast the tangent lines change direction.
- Lecture Videos:
- Learn and apply the Squeeze Theorem to limits of functions which are easy to bound.
- Recognize limit problems which can be solved by utilizing the new limit identities sin(x)/x and (cos(x)-1)/x as x approaches 0.
- Learn the derivatives of the six trigonometric functions and combine these forms with the differentiation rules developed in previous sections.
- Learn how to find the derivative of inverse functions using implicit differentiation; in particular, learn the derivatives and their domain for the arctangent, arcsine, and arccosine functions.
- Lecture Videos:
- More Videos on Trigonometric Derivatives:
- Become familiar with the Chain Rule in both the Lagrange and Leibniz form.
- Master the use of the Chain Rule with the other differentiation rules concerning arithmetic operations of functions.
- Understand how to iterate the Chain Rule when calculating the derivative of functions which can be written as multiple function compositions.
- Lecture Videos:
- Identify functions as either implicitly or explicitly presented.
- Learn how to differentiate implicit functions by extending the rules of differentiation for explicit functions developed in previous sections.
- Interpret derivatives with respect to a variable in the context of examples.
- Lecture Videos:
- Learn how to find the derivative of logarithmic functions and combine these forms with the derivative rules developed in previous sections.
- Understand and be able to identify when logarithmic differentiation can be used or useful.
- Lecture Videos:
- Interpret word problems by utilizing diagrams, identifying relevant variables, and constructing equations to represent relationships between variables.
- Demonstrate mastery of basic differential calculus by identifying and calculating rates described by word problems.
- Lecture Videos:
- Utilize linearization to estimate the value of a differentiable function.
- Utilize differentials to approximate change; in particular, use differentials to estimate the error of a measurement.
- Lecture Videos:
- Identify local and absolute extrema on a graph. Understand the difference between an extrema value and an extreme point.
- Demonstrate understanding of Fermat's Theorem and critical values by computing possible local extrema.
- Demonstrate understanding of the Extreme Value Theorem by utilizing the Closed Interval Method to find absolute extrema.
- Lecture Videos:
- Understand the Mean Value Theorem and identify its uses.
- Utilize the First Derivative Test to identify local extrema; in particular, describe the shape of the graph of a function using intervals on which the function increases or decreases.
- Lecture Videos:
- Utilize the Second Derivative Test to identify local extrema.
- List the strengths and weaknesses, with explicit examples, of the Second Derivative Test.
- Describe the shape of the graph of a function using intervals on which the function increases or decreases and intervals on which the function is concave up or down.
- Lecture Videos:
- Be able to list the indeterminate forms.
- Identify and evaluate limits which satisfy the requirements of l'Hospital's Rule.
- Master the techniques which allow for the evaluation of limits with indeterminate products and powers using l'Hospital's Rule.
- Lecture Videos:
- Using intervals of increasing decreasing, concavity, extrema, end behavior, asymptotes and sample points to graph a function.
- Lecture Videos:
- Apply the Closed Interval Method and the First/Second Derivative Tests to word problems to find an optimal solution.
- Lecture Videos:
- Learn Newton's Method and use it to approximate the roots of functions.
- Lecture Videos:
- Approximate area by tiling a region with rectangles.
- Demonstrate understanding that distance is related to area by estimating distance displaced using the area under the curve of the velocity function.
- Learn and become familiar with the sigma notation for summation.
- Lecture Videos:
- Definite integrals are defined as a limit of Riemann Sums. Be able to convert a definite integral into a limit of Riemann Sums and vice versa.
- Demonstrate understanding that definite integrals represent signed area by utilizing the 8 integral properties derived from this connection.
- Lecture Videos:
- Understand that indefinite integrals are general antiderivatives of their integrand.
- Understand that antiderivatives are not unique; be able to describe this using the Mean Value Theorem.
- Demonstrate mastery of antiderivatives by calculating general antiderivatives and identifying antiderivatives in problems and graphs.
- Lecture Videos:
- Demonstrate understanding of the Fundamental Theorem of Calculus I by utilizing it in integrating continuous functions.
- Lecture Videos:
- Demonstrate understanding of the Fundamental Theorem of Calculus II by utilizing it in differentiating integral functions.
- Understand that a major consequence of the Fundamental Theorem of Calculus II is that every continuous function has an antiderivative.
- Lecture Videos:
- Demonstrate understanding of the Substitution Method by successfully changing the variable of integration and evaluating the resulting integral.
- Lecture Videos:
|
Using Basic Limit Laws |
# 1-30 |
|
Complexities and Counterexamples |
# 31-35 |
The Basic Laws of Limit
Assumptions Matters
Section 2.4: Limits and Continuity.
Please print the prelecture notes before the lecture: Section 2.4 PreLecture (Section 2.4 PostLecture)Learning objectives are
|
Complexities and Counterexamples |
# 1-6, 62-66, 81-84 |
|
Identifying Discontinuities |
# 17-34, 49 |
|
Discontinuities of Piecewise Functions |
# 51-55, 57-60 |
|
Evaluating Limits by Substitution |
# 67-80 |
Continuity and Discontinuity
Continuity and Elementary Functions
Section 2.5: Indeterminate Forms.
Please print the prelecture notes before the lecture: Section 2.5 PreLecture (Section 2.5 PostLecture)Learning objectives are
|
Limit Calculation Techniques |
# 5-34, 43-54 |
Conjugation and Simplification
The Squeeze Theorem
Piecewise-defined
Section 2.6: The Squeeze Theorem and Trigonometric Limits.
Please print the prelecture notes before the lecture: Section 3.6 PreLecture (Section 3.6 PostLecture)Learning objectives are
Section 2.7: Limits at Infinity.
Please print the prelecture notes before the lecture: Section 2.7 PreLecture (Section 2.7 PostLecture)Learning objectives are
|
Evaluating Limits at Infinity Graphically |
# 1-4, 35-42 |
|
Evaluating Limits at Infinity Algebraically |
# 7-30, 33, 43 |
Graphs at Infinity, HA, VA
Graphs at Infinity and Composition
Section 2.8: The Intermediate Value Theorem.
Please print the prelecture notes before the lecture: Section 2.8 PreLecture (Section 2.8 PostLecture)Learning objectives are
|
Using the Theorem |
# 1-17 |
|
Does the Theorem Apply? |
# 18-20 |
|
The Bisection Method |
# 21-24 |
|
Creating Examples |
# 25-30 |
Intermediate Value Theorem
Bisection Method
Sections 3.1: Definition of the Derivative.
Please print the prelecture notes before the lecture: Section 3.1 PreLecture (Section 3.1 PostLecture)Geogebra Sheet for Oscillating Functions
Learning objectives are
|
Computing Derivatives |
# 3-8, 28-46 |
|
Derivatives and Graphs |
# 13-16, 47, 48 |
|
Identifying Derivatives from Limits |
# 53-58 |
Slope of tangent line
Continuity & Non-diff. Graphs
Sections 3.2: The Derivative as a Function.
Please print the prelecture notes before the lecture: Section 3.2 PreLecture (Section 3.2 PostLecture)Learning objectives are
|
Computing Derivatives with Limits |
# 1-6 |
|
Computing Derivatives with Rules |
# 7-32, 48-49 |
|
Concepts and Complexities |
# 43-47, 50-56, 66, 68 |
Notation of Derivative
Motion & Derivative Rules
Derivative of the Expotential Function
Using the Derivative Function
Sections 3.3: Product and Quotient Rules.
Please print the prelecture notes before the lecture: Section 3.3 PreLecture (Section 3.3 PostLecture)Learning objectives are
|
Calculating Derivatives with Rules |
# 1-42 |
|
Concepts and Complexities |
# 51, 54-56 |
Product Rule
Quotient Rule
Tabular Examples
Sections 3.4: Rates of Change.
Please print the prelecture notes before the lecture: Section 3.4 PreLecture (Section 3.4 PostLecture)Learning objectives are
|
Rates of Change |
# 1-11, 18-23, 25-29 |
|
Estimating Change |
# 39-45 |
1-unit Change
Kinematics
Other Applications
Sections 3.5: Higher Derivatives.
Please print the prelecture notes before lectureSection 3.5 PreLecture (Section 3.5 PostLecture)Learning objectives are
|
Calculating Higher Derivatives |
# 1-38 |
|
Graphing Higher Derivatives |
# 39, 40 |
Lagrange & Leibneiz Notation
Increase/Decrease & Concavity
Higher Derivatives & Implicit Diff.
Sections 3.6: Trigonometric Functions.
Please print the prelecture notes before the lecture: Section 3.6 PreLecture (Section 3.6 PostLecture)Learning objectives are
|
Derivatives of Trigonometric Functions |
Page 158 # 5-24, 39-45 |
|
Derivatives of Inverse Trigonometric Functions
|
Page 172 # 27-44 |
The Inequality (proof)
The trigonemetric Limits
Derivative of Other Trig Functions
A Related Rate Problem
Trigonometric Function Limits
Trigonometric Function Limits
Derivative of Arc-functions
Sections 3.7: The Chain Rule.
Please print the prelecture notes before the lecture: Section 3.7 PreLecture (Section 3.7 PostLecture)Learning objectives are
|
Calculating Derivatives with Rules (for certain problems, look up trig derivatives) |
# 1-76 |
|
Concepts and Complexities |
# 77-80 |
Composite Functions
The Chain Rule
Implicit Differentiation
The Generalized Power Rule
Sections 3.8: Implicit Differentiation.
Please print the prelecture notes before lectureSection 3.8 PreLecture (Section 3.8 PostLecture)Learning objectives are
|
Implicit Differentiation: Calculations |
# 1-26 |
|
Implicit Differentiation: Tangent Lines |
# 53-64 |
Implicit Functions
Implicit Differentiation
Derivative of Logarithmic Functions
Shadowing Related Rates
Sections 3.9: Derivatives of General Exponential and Logarithmic Functions.
Please print the prelecture notes before the lecture: Section 3.9 PreLecture (Section 3.9 PostLecture)Learning objectives are
|
Logarithmic Differentiation |
# 1-50, 78-80 |
Logarithmic Laws & Derivatives
Steps
Implicit Logarithmic Differentiation
Sections 3.10: Related Rates.
Please print the prelecture notes before the lecture: Section 3.10 PreLecture (Section 3.10 PostLecture)Learning objectives are
|
Related Rates Word Problems |
# 1-39 |
Introduction
Examples
Some Trig Examples
Similar Triangle Examples
Conical Reservoir
Clock Example
iClicker
Arrow Example
Sections 4.1: Linear Approximation and Applications.
Please print the prelecture notes before the lecture: Section 4.1 PreLecture (Section 4.1 PostLecture)Learning objectives are
|
Approximating Differences with Tangent Lines |
# 1-24 |
|
Word Problems |
# 27-33, 41-43 |
|
Linearization |
# 45-57 |
Linearization
Differentials
Relative Error
Sections 4.2: Extreme Values.
Please print the prelecture notes before the lecture: Section 4.2 PreLecture (Section 4.2 PostLecture)Learning objectives are
|
Finding Critical Points |
# 1-24 |
|
The Closed Interval Method |
# 29-58 |
|
Rolle's Theorem |
# 65-72 |
|
Concepts and Complexities |
# 86-90 |
Absolute and Local Extremum
Fermat & Extreme Value Theorem
The Closed Interval Method
Closed Interval Method Examples
Sections 4.3: The Mean Value Theorem and Monotonicity.
Please print the prelecture notes before the lecture: Section 4.3 PreLecture (Section 4.3 PostLecture)Learning objectives are
|
Mean Value Theorem Applications |
# 1-12, 64, 68-71 |
|
Concepts and Complexities |
# 15-22 |
|
Intervals on which a Function Increases or Decreases |
# 27-62, 65 |
Roll's Theorem
The Mean Value Theorem
Some Proofs
First Derivative Test
Sections 4.4: The Second Derivative and Concavity.
Please print the prelecture notes before the lecture: Section 4.4 PreLecture (Section 4.4 PostLecture)Learning objectives are
|
Concepts and Complexities |
# 1-8, 51-54 |
|
Curve Sketching |
# 9-38 |
Concavity
Second Derivative Test
More Second Derivative Test
Sections 4.5: L’Hôpital’s Rule.
Please print the prelecture notes before the lecture: Section 4.5 PreLecture (Section 4.5 PostLecture)Learning objectives are
|
Evaluating Limits with L'Hôpital's Rule |
# 1-54, 57-59, 61 |
|
Comparable Growth |
# 64-67 |
Indeterminate Forms & Theorem
Using LHR Twice
Forms (∞ - ∞) and (0 ✗ ∞)
Form 1∞
Rate of Growth
Sections 4.6: Analyzing and Sketching Graphs of Functions.
Please print the prelecture notes before the lecture: Section 4.6 PreLecture (Section 4.6 PostLecture)Learning objectives are
|
Concepts and Complexities |
# 1-8, 51-54 |
|
Curve Sketching |
# 9-38 |
Introduction
A Rational Example
A Radical Example
A Logarithmic Example
An Exponential Example
Examples with Unknown Parameters
Graph Using derivatives Only
Sections 4.7: Applied Optimization.
Please print the prelecture notes before the lecture: Section 4.7 PreLecture (Section 4.7 PostLecture)Learning objectives are
|
Optimization Word Problems |
# 1-70 |
Introduction
Examples, Closed Interval Method
More Quadratic Examples
Open Box Example
Two Rate Optimization
Landscaping Example
Moving Objects & Embedding Circles
The Gutter Example
Sections 4.8: Newton’s Method.
Please print the prelecture notes before the lecture: Section 4.8 PreLecture (Section 4.8 PostLecture)Learning objectives are
|
Applying Newton's Method |
# 1-15 |
|
Concepts and Complexities |
# 27-30 |
Video is coming in spring
Newton'S Method
Video is coming in spring
Examples
Video is coming in spring
Multiple Roots
Sections 5.1: Approximating and Computing Area.
Please print the prelecture notes before the lecture: Section 5.1 PreLecture (Section 5.1 PostLecture)Learning objectives are
|
Approximating Area with Rectangles |
# 1-6, 12-14 |
|
Summation Notation |
# 23-32, 63-66 |
Introduction & Reimman Sums
Summation Notation
The Distance and Area
Sections 5.2: The Definite Integral.
Please print the prelecture notes before the lecture: Section 5.2 PreLecture (Section 5.2 PostLecture)Learning objectives are
|
Evaluating Integrals with Geometry |
# 1-10, 13-16, 71-74 |
|
Areas of Positive and Negative Values |
# 23-28 |
|
Evaluating Integrals using Basic Properties |
# 33-66, 75-82 |
The Area
The Properties
The Comparison Properties
Sections 5.3: The Indefinite Integral.
Please print the prelecture notes before the lecture: Section 5.3 PreLecture (Section 5.3 PostLecture)Learning objectives are
|
General Antiderivatives |
# 1-39, 42-46 |
|
Antiderivatives: Graphs |
# 40, 41 |
|
Antiderivatives: Initial Value Problems |
# 47-69 |
|
Concepts and Complexities |
# 70-80 |
The General Antiderivative
Examples
Application
More Applications
Sections 5.4: The Fundamental Theorem of Calculus, Part I.
Please print the prelecture notes before the lecture: Section 5.4 PreLecture (Section 5.4 PostLecture)Learning objectives are
|
Evaluating Integrals with FTCI |
# 1-42, 49-57 |
|
FTCI and Absolute Values |
# 43-48 |
|
Concepts and Complexities |
# 49-57, 62 |
The Fundamental Theorem
The Fundamental Theorem
The Net Change
Total Distance
Sections 5.5: The Fundamental Theorem of Calculus, Part II.
Please print the prelecture notes before the lecture: Section 5.5 PreLecture (Section 5.5 PostLecture)Learning objectives are
|
The Area Function |
# 1-6, 21-26, 29-39, 47 |
The Cumulative Function
Examples
Sections 5.7: The Substitution Method.
Please print the prelecture notes before the lecture: Section 5.7 Pre and PostLectureLearning objectives are
|
Integration: Substitution Method |
# 1-98, 101-104 |