Review (with solutions and in class slides)

All Review Videos and Gateway Videos PlayList (Click)

Worksheets (Solutions are not posted here. If you are an instructor, feel free to contact me)

Disclaimer: Much of this content was created by Dr. Joe Brennan or in collaboration with him. Professor Jeremy Martin has contributed to the slides in Spring 2018. Among other content, almost all of the tikz content and LaTeX Solutions are made by me. Many of our wonderful graduate students have contributed to the course by proofreading and giving feedback; I appreciate all of them. The original idea for the template of schedules came from Dr. Sefika Kuzgun. If you are using any of this content, please give credit to the persons it belonged to.

If you are an instructor and wish to use the solutions, rubrics and TA instructions of the worksheets, feel free to contact me.

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Sylabus and Schedule

Math125-Syllabus.pdf

Math125-Schedule.pdf

Review (with solutions and in class slides)

Math_125_Review 1.pdf

Math_Review1_Solution.pdf

Math_125_Review 2.pdf

Main Review 2 Solution Math 125.pdf

Math_125_Main_Final_Review.pdf

Math125-Main Final Review Solution.pdf

Review 2 in Lecture Slides Math 125.pdf

Math_125_Review 2_Slides.pdf

Math_125_Main Final Review Slides.pdf

Slide_125_Final Review.pdf

Practice Exams and Solutions

Math125-Practice 1.pdf

Math 125- Practice 1 Solution.pdf

Math125-Practice 2.pdf

Math125-Practice 2 -Solution.pdf

Math125-Quiz4-Practice Solution.pdf

Math125_FInal_Practice.pdf

Math125-Final Practice Solution.pdf

All Review Videos and Gateway Videos PlayList (Click)

Precalculus Review: https://www.youtube.com/playlist?list=PLdAr7Wo2nI-22kgOzkJDQXYQti5WY1NJl
Gateway: https://www.youtube.com/playlist?list=PLdAr7Wo2nI-25-4TFOY1DjgjQXDPRTpLC
Midterm 1: https://www.youtube.com/playlist?list=PLdAr7Wo2nI-2gkkMAa4Q2036Loo27Efar
Midterm 2: https://www.youtube.com/playlist?list=PLdAr7Wo2nI-0WAOiWiuETe5jFDt7JQDMh
Final: https://www.youtube.com/playlist?list=PLdAr7Wo2nI-1EEh5TopgX2NYfw8Dq4DT0

Gateway Pools of Questions

Math125_All gateway Pools.pdf

Pre and Post Lecture Notes

Chapter 2: Limits

Videos: https://www.youtube.com/playlist?list=PLdAr7Wo2nI-1TW63kmZu8k4fi4SjoAeLw

More Videos: https://www.youtube.com/playlist?list=PLdAr7Wo2nI-2owPkUfgwvp9qKjgEO07r_

Learning objectives are

Section 2.1

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.

Section 2.2

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.

Section 2.3

Identify functions and values for which the basic limit laws apply.
Use the five basic limit laws to evaluate limits.

Section 2.4

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.

Section 2.5

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.

Section 2.7

Learn the term asymptote and master identifying both vertical and horizontal asymptotes graphically and algebraically.
Learn the asymptotes of common functions.

Section 2.8

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.

Math_125_First Day Review Post.pdf

Math_125_First Day Review Pre.pdf

Math_125_Section 2.1 Post.pdf

Math_125_Section 2.1 Pre.pdf

Math_125_Section 2.2 Post.pdf

Math_125_Section 2.2 Pre.pdf

Math_125_Section 2.3 Pre.pdf

Math_125_Section 2.3 Post.pdf

Math_125_Section 2.4 Pre.pdf

Math_125_Section 2.4 Post.pdf

Math_125_Section 2.5 Pre.pdf

Math_125_Section 2.5 Post.pdf

Math_125_Section 2.7 Post.pdf

Math_125_Section 2.7 Pre.pdf

Math_125_Section 2.8 Pre.pdf

Math_125_Section 2.8 Post.pdf

Chapter 3: Derivatives

Oscilating derivatives

Learning objectives are

Section 3.1

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.

Section 3.2

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.

Section 3.3

Master the Product and Quotient Rules of Differentiation; in particular, learn how to combine these rules with those developed in section 3.2.

Section 3.4

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.

Section 3.5

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.

Section 2.6 and 3.6

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.

Section 3.7

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.

Section 3.8

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.

Section 3.9

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.

Section 3.10

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.

Math_125_Section 3.1 Pre.pdf

Math_125_Section 3.1 Post.pdf

Math_125_Section 3.2 Post.pdf

Math_125_Section 3.2 Pre.pdf

Math_125_Section 3.3 Post.pdf

Math_125_Section 3.3 Pre.pdf

Math_125_Section 3.4 Post.pdf

Math_125_Section 3.4 Pre.pdf

Math_125_Section 3.5 Pre.pdf

Math_125_Section 3.5 Post.pdf

Math_125_Section 3.6 Post.pdf

Math_125_Section 3.6 Pre.pdf

Math_125_Section 3.7 Post.pdf

Math_125_Section 3.7 Pre.pdf

Math_125_Section 3.8 Pre.pdf

Math_125_Section 3.8 Postpdf.pdf

Math_125_Section 3.9 Post.pdf

Math_125_Section 3.9 Pre.pdf

Math_125_Section 3.10 Pre.pdf

Math_125_Section 3.10 Post.pdf

Chapter 4: Application of Derivatives

Section 4.1

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.

Section 4.2

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.

Section 4.3

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.

Section 4.4

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.

Section 4.5

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.

Section 4.6

Using intervals of increasing decreasing, concavity, extrema, end behavior, asymptotes and sample points to graph a function.

Section 4.7

Apply the Closed Interval Method and the First/Second Derivative Tests to word problems to find an optimal solution.

Section 4.8

Learn Newton's Method and use it to approximate the roots of functions.

Math_125_Section 4.1 Post.pdf

Math_125_Section 4.1 Pre.pdf

Math_125_Section 4.2 Pre.pdf

Math_125_Section 4.2 Post.pdf

Math_125_Section 4.3 Post.pdf

Math_125_Section 4.3 Pre.pdf

Math_125_Section 4.4 Post.pdf

Math_125_Section 4.4 Pre.pdf

Math_125_Section 4.5 Pre.pdf

Math_125_Section 4.5 Post.pdf

Math_125_Section 4.6 Pre.pdf

Math_125_Section 4.6 Post.pdf

Math_125_Section 4.7 Post.pdf

Math_125_Section 4.7 Pre.pdf

Math_125_Section 4.8 Pre.pdf

Math_125_Section 4.8 Post.pdf

Chapter 5: Integration

Learning objectives are

Section 5.1

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.

Section 5.2

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.

Section 5.3

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.

Section 5.5

Demonstrate understanding of the Fundamental Theorem of Calculus I by utilizing it in integrating continuous functions.

Section 5.7

Demonstrate understanding of the Substitution Method by successfully changing the variable of integration and evaluating the resulting integral.

Math_125_Section 5.1 Pre.pdf

Math_125_Section 5.1 Post.pdf

Math_125_Section 5.2 Post.pdf

Math_125_Section 5.2 Pre.pdf

Math_125_Section 5.3 Pre.pdf

Math_125_Section 5.3 Post.pdf

Math_125_Section 5.4 Post.pdf

Math_125_Section 5.4 Pre.pdf

Math_125_Section 5.5Pre.pdf

Math_125_Section 5.5Post.pdf

Math_125_Section 5.7 Post.pdf

Math_125_Section 5.7 Pre.pdf

Worksheets (Solutions are not posted here. If you are an instructor, feel free to contact me)

WS1.pdf

WS2.pdf

WS10.pdf

WS3.pdf

WS4.pdf

WS5.pdf

WS6.pdf

WS7.pdf

WS8.pdf

WS9.pdf

WS10.pdf

WS11-MATH-2WQ0AXJ1G9.pdf

WS12.pdf

WS13.pdf

WS14.pdf

WS15.pdf

WS16.pdf

WS17.pdf

WS18.pdf

WS19.pdf

WS20.pdf

WS21.pdf