MAT2410: Calculus I

Textbook: Stewart J 2016. Calculus, Eighth Edition, Boston, MA: Cengage Learning, ISBN: 978-1-285-74062-1

chapter 1 functions and limits

1.1 four ways to represent a function

1.2 mathematical models: a catalog of essential functions

1.3 new functions from old functions

1.4 tangent and velocity problems

1.5 the limit of a function

Objectives

The learners will...

Explore an intuitive definition of the limit
Investigate limits numerically
Investigate limits graphically
Explore infinite limits

The following video describes how to find limits numerically.

https://www.youtube.com/watch?v=t7NvlTgMsO8

This video goes through the evaluation of limits from a graph.

https://www.youtube.com/watch?v=3YFuOwcog6k

1.6 calculating limits using the limit laws

Objectives

The learners will...

Describe and utilize the basic Laws of Limits

This video describes the evaluation of basic limits.

https://www.youtube.com/watch?v=CLu5CIR7uyc

Limits can be evaluated analytically.

https://www.youtube.com/watch?v=VLiMfJHZIpk

Limits can be found by rationalizing a denominator. This is example 1.

https://www.youtube.com/watch?v=ptgGkxhExA0

Limits can be found by rationalizing a denominator. This is example 2.

https://www.youtube.com/watch?v=-zE63L4AA94

Finding the limit of a rational function (Example 1).

https://www.youtube.com/watch?v=5q8KMcQYnvE

Finding the limit of a rational function (Example 2).

https://www.youtube.com/watch?v=qHfyB0J57qo

Finding the limit of a rational function (Example 3).

https://www.youtube.com/watch?v=gRk24f3SUWQ

Finding the limit of a rational function (Example 4).

https://www.youtube.com/watch?v=ag7mi0GL6PM

Finding the limit of a rational function (Example 5).

https://www.youtube.com/watch?v=Omeiuy7jclA

Finding the limit of a function containing an absolute value.

https://www.youtube.com/watch?v=AQTdZjT0N54

Finding the limit of a piecewise function.

https://www.youtube.com/watch?v=8ZOXR4pn3Qg

1.7 the precise definition of a limit

Objectives

The learners will...

Learn the terminology and procedure of precisely proving a limit

This describes the formal definition of a limit (part 1).

https://www.youtube.com/watch?v=NAAhJ5ucYJg

This describes the formal definition of a limit (part 2).

https://www.youtube.com/watch?v=ccaUP7d7hAo

This focuses on finding delta values, given an epsilon.

https://www.youtube.com/watch?v=EHdlietoEf8

Finding delta values, given an arbitrary epsilon (example 1).

https://www.youtube.com/watch?v=nfZeOCmIRbU

Finding delta values, given an arbitrary epsilon (example 2).

https://www.youtube.com/watch?v=5Lfa237g4NY

1.8 continuity

Objectives

The learners will...

Define and determine continuity at a point
Define and determine continuity from the right or left
Define and determine continuity on an interval
Define and utilize the Intermediate Value Theorem

chapter 2 derivatives

2.1 derivatives and rates of change

Objectives

The learners will...

Define the derivative and understand its multiple interpretations
Calculate a derivative using limits

The derivative as a rate of change.

https://www.youtube.com/watch?v=uePqXn7FXvY

Interpreting the derivative - Marginal cost

https://www.youtube.com/watch?v=grS6HP80V4A

Interpreting the derivative - Population growth

https://www.youtube.com/watch?v=UFZf4gIsaOA

Interpreting the derivative - Fluctuations in Temperature

https://www.youtube.com/watch?v=XUVXfD8zA-s

2.2 the derivative as a function

Objectives

The learners will...

Calculate the derivative function using limits
Define differentiability
Calculate higher-order derivative functions

Determine the derivative at a point using the limit definition - Example 1

https://www.youtube.com/watch?v=9ZYNJN_yMMk

Determine the derivative at a point using the limit definition - Example 2

https://www.youtube.com/watch?v=NY9lh4vclEk

Determine the derivative of a function using the limit definition - Example 1

https://www.youtube.com/watch?v=BRm2L2a9BTs

Determine the derivative of a function using the limit definition - Example 2

https://www.youtube.com/watch?v=1a9yyxvSfso

Determine the derivative of a function using the limit definition - Example 3

https://www.youtube.com/watch?v=569TJtZD7NE

Determine the derivative of a function using the limit definition - Example 4

https://www.youtube.com/watch?v=VqWixrdIe9A

Sketch the graph of the derivative based upon a graph of the function.

https://www.youtube.com/watch?v=syqk5Rif9AU

2.3 differentiation rules

Objectives

The learners will...

Derive and utilize formulas for differentiation of multiple different function types

The constant rule, the constant multiple rule, the sum and different rule, and the power rule.

https://www.youtube.com/watch?v=e1GMC9aOyBU

https://www.youtube.com/watch?v=cEvhWaQZwGA

Finding derivatives using the power rule - multiple examples

https://www.youtube.com/watch?v=QJBV7LQ2T7c

https://www.youtube.com/watch?v=w-AV_DjFdbk

Using the power rule with radicals or rational or negative exponents - multiple examples

https://www.youtube.com/watch?v=OqsVPz8I-_Y

https://www.youtube.com/watch?v=nPY2lRa8_V0

https://www.youtube.com/watch?v=fHbIPcrmgfk

https://www.youtube.com/watch?v=WXnmXEeHhLM

https://www.youtube.com/watch?v=5KZsvwZkgSw

https://www.youtube.com/watch?v=b0_OHsMj12g

Find derivatives using the product rule - multiple examples

https://www.youtube.com/watch?v=3T0mD8vKLN4

https://www.youtube.com/watch?v=gsaHoc2mSps

https://www.youtube.com/watch?v=baHNv3Ay8ls

https://www.youtube.com/watch?v=jsGwuXB28F0

https://www.youtube.com/watch?v=RBiOxTj_lZI

Find derivatives using the quotient rule - multiple examples

https://www.youtube.com/watch?v=wg2O3o4s1q0

https://www.youtube.com/watch?v=Inq4V8Vogxc

https://www.youtube.com/watch?v=u8a8ShW2iMk

https://www.youtube.com/watch?v=RhtQSxuYMac

https://www.youtube.com/watch?v=jrpnmFH5-ls

https://www.youtube.com/watch?v=ekztH8WLu5A

https://www.youtube.com/watch?v=kE_dV2qvTng

Using the quotient rule with a graph.

https://www.youtube.com/watch?v=W0gdymaqovg

Finding velocity and acceleration.

https://www.youtube.com/watch?v=-xyOdqYS6-A

2.4 derivative of trigonometric functions

Objectives

The learners will...

Derive and utilize some important limits involving trigonometric functions
Derive and utilize derivatives of trigonometric functions

The derivatives of sine and cosine.

https://www.youtube.com/watch?v=aFcYYT26tCo

Using the product and quotient rules with trigonometric functions

https://www.youtube.com/watch?v=0nG-wx2nkDI

Derivatives and derivative values of basic sine and cosine functions

https://www.youtube.com/watch?v=NG-P2Jxxg0c

Finding derivatives and equations of tangent lines for trigonometric functions

https://www.youtube.com/watch?v=k5gxVoXtB-k

Derivatives and derivative values of cosine and cosecant functions

https://www.youtube.com/watch?v=olr3VF9qxPU

2.5 the chain rule

Objectives

The learners will...

Utilize the Chain Rule in a variety of contexts

The chain rule - parts 1 and 2

https://www.youtube.com/watch?v=PT11CSo-WkQ

https://www.youtube.com/watch?v=1x_wMYFOliQ

The chain rule - concept checks

https://www.youtube.com/watch?v=zrkJyBjH_iA

https://www.youtube.com/watch?v=efXyBRNhnG4

Examples using the chain rule ---

Quadratic raised to a power

https://www.youtube.com/watch?v=9PbGK4IxB-M

Negative exponent

https://www.youtube.com/watch?v=flMCWyTiQGs

Polynomial to a power

https://www.youtube.com/watch?v=4sqt5Z2EDLs

Polynomial inside a square root

https://www.youtube.com/watch?v=gzal2-OvEwk

Polynomial in a denominator raised to a power

https://www.youtube.com/watch?v=TjVC228mDk4

Trigonometric function raised to a power

https://www.youtube.com/watch?v=uAPuNMUDQkk

https://www.youtube.com/watch?v=OEvH9gxCT7w

The chain rule with the product rule

https://www.youtube.com/watch?v=rjDPc8oHtZI

The chain rule with the quotient rule

https://www.youtube.com/watch?v=IHENYo3ogv8

Using the chain rule twice

https://www.youtube.com/watch?v=AcEvUtzqq1Y

2.6 implicit differentiation

Objectives

The learners will...

Employ the derivative as an operator
Implicitly differentiate an equation
Find dy/dx by implicit differentiation

Introduction to Implicit Differentiation

https://www.youtube.com/watch?v=1DrPwAyAhsE

https://www.youtube.com/watch?v=1scXr6g7HdA

Implicit Differentiation - Examples

https://www.youtube.com/watch?v=kroPpRJI0QM

https://www.youtube.com/watch?v=oUQDDiPOLNs

https://www.youtube.com/watch?v=8VQVErQmOa0

https://www.youtube.com/watch?v=5hjA_Ty_e3U

2.7 rates of change in the natural and social sciences

Objectives

The learners will...

Explore applications of derivatives in multiple scientific fields

2.8 related rates

Objectives:

The learners will...

Apply derivatives to real-world problems involving changes in multiple related variables

The ball volume problem

https://www.youtube.com/watch?v=zXrm-Smy0qU

The ladder problem

https://www.youtube.com/watch?v=ZAgFwsOxSgU

The right circular cone problem

https://www.youtube.com/watch?v=sfS7VfJywPo

Rotating light projecting on a wall

https://www.youtube.com/watch?v=zIqJnkGCVDw

Air volume and pressure

https://www.youtube.com/watch?v=siQdiFGseJ0

Rate of change of shadow from a light pole

https://www.youtube.com/watch?v=mejD8lu4sNc

2.9 linear approximations and differentials

Objectives:

The learners will...

Create linearization approximations to functions at a point
Create differentials of functions and utilize them to solve problems

chapter 3 applications of differentiation

3.1 maximum and minimum values

Objectives

The learners will...

Use the derivative to find function maxima and minima

3.2 the mean value theorem

3.3 how derivatives affect the shape of a graph

3.4 limits at infinity - horizontal asymptotes

3.5 summary of curve sketching

3.7 optimization problems

3.a derivatives of exponential and logarithmic functions

3.b l'hopital's rule

3.8 newton's method

3.9 antiderivatives

chapter 4 integrals

4.1 areas and distances

4.2 the definite integral

4.3 the fundamental theorem of calculus

4.4 integration formulas and the net change theorem

4.5 the substitution rule

chapter 5 applications of integration

5.1 areas between curves

5.2 volumes

5.3 volumes by cylindrical shells

5.4 work, energy, and averages

chapter 6 inverse functions

6.1 inverse functions

6.2-6.4 integrals, exponential functions, and logarithms

6.5 exponential growth and decay

6.6 integrals resulting in inverse trigonometric functions

6.7 hyperbolic functions

chapter 7 techniques of integration

7.1 integration by parts

7.2 trigonometric integrals

7.3 trigonometric substitution

7.4 partial fractions

7.7 approximate integration

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