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College Calculus: Level I Review of Functions

Section 1: Overview of Functions: Lecture 1 | 26:29 min

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Lecture Comments (22)

	1 answer Last reply by: Dr. Jennifer Switkes Mon Nov 12, 2018 6:57 PM Post by Karoly Ficsor on August 5, 2018 why is it not possible to view Additional Examples 3 and 4?
	0 answers Post by Mohamed E Sowaileh on July 10, 2017 Hello Dr. Jenny Switkes, I hope you are very well. I am a student who is extremely weak in math. In order to be very strong in math, specially for engineering field, could you provide me with sequential order of mathematical topics and textbooks. With what should I begin so that I can master big topics like calculus, statistics, probability ... etc. Your guidance is precious to me. Thank you so much.
	1 answer Last reply by: Firebird wang Wed Nov 2, 2016 9:55 PM Post by MOHAMMED ALHUMAIDI on June 16, 2016 Please where can I find the downloadable lecture slides?
	0 answers Post by Daniel Gonzalez on June 1, 2016 where can I find the downloadable lecture slides?
	1 answer Last reply by: Bilbo Baggins Sat Sep 5, 2015 4:19 PM Post by enya zh on August 19, 2015 You are soooooooooooo much better than Zhu(the person who teaches AP Calculus AB & BC).â˜ºâ˜ºâ˜ºâ˜ºâ˜ºâ˜ºâ˜ºâ˜º
	0 answers Post by Matthew Favazza on January 21, 2015 The watch clinging the screen is very, very distracting...
	0 answers Post by Timothy Davis on August 10, 2014 Hello Professor Switkes Wouldn't sin(-x) = -sin(-x)? Isn't the definition of an odd function f(x) = -f(-x)?
	1 answer Last reply by: Hee Su Jang Mon Jul 14, 2014 6:30 PM Post by Taylor Wright on August 7, 2013 At 12:30 Why would it not include zero? if x=0, then the sqrt of zero is zero, which would be included in the function.
	1 answer Last reply by: Taylor Wright Wed Aug 7, 2013 1:49 AM Post by Maureen Dempsey on March 27, 2013 hi what is the difference between exponenial and power function....don't they both involve an exponent? She didn't really clarify how they are different. thanks for a great tutoria otherwise.
	0 answers Post by Eun Jee Kang on October 9, 2012 I can't continue the lesson after odd and even examples. Please check it out. i don't know what problem is.
	0 answers Post by Jacob Mack on August 8, 2012 y = x^2 + 1 would be a function but y^2 = x + 1 is not. To see this algebraically instead of geometrically we can plug a number in for x for the first equation, say, 2, so 2 squared is four + 1 is 5. We have exactly one value of x domain we get one output of y in the equation. Thus, 2^2 + 1 = 5 is a function. Plugging in 3 for the second equation we see it is not a function: y^2 = 3 + so y can be = 2 or -2, therefore there are two outputs from the range y of the equation.
	0 answers Post by Jacob Mack on August 8, 2012 In general, any equation can yield a corresponding graph and any graph can be represented by a corresponding equation. Like y = x^2 yields the familiar parabola. For y = log(x) we can break it apart to be y = f(x) = 1n(x) + x and then move on to solve geometrically.
	0 answers Post by Jason Mannion on October 4, 2011 Video works fine for me so far. Hope to get a better handle on Calculus now!
	0 answers Post by Marsha Taylor on January 23, 2011 These videos are a great for learning math or for review.
	0 answers Post by NICK FOSTER on January 20, 2011 no iam sorry about that comment everything is ok the videos play and are amazing!!keep up the good job

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Review of Functions

These are review topics from algebra and pre-calculus – it would be good to just briefly “brush up” on these things!
As you go through calculus, it will be important to use the correct terminology for the various terms associated with functions – clear mathematical communication is important!

Review of Functions

Simplify tanx cscx

y = tanx cscx
= [sinx/cosx] [1/sinx]

= [1/cosx] = secx

Simplify cscx cosx

y = cscx cosx
= [1/sinx] cosx
= [cosx/sinx]

=cotx

Simplify [4x sinx/(x² cos3x)]

y = [4x sinx/(x² cos3x)]
= [4x/(x²)] [sinx/cos3x]

and cannot be combined because their arguments differ. The arguments must be the same to rewrite trig functions. 0Find the domain and range of

sin and cos are periodic functions, meaning the outputs will repeat once the input is shifted over a full period. The unit circle is a good way to see this. No matter what real numbers we put into the function, the same pattern repeats. The output never exceeds 1 or goes below -1

Domain (), Range [-1, 1] 0Find the domain and range of

This is similar to the previous problem, but it's a bit trickier because it introduces zero into the denominator. The function is said to be undefined where the input results in a zero in the denominator. For this function, we won't have any problems when the denominator is not zero. So the domain involves any x when is not zero. is zero at any multiple of , including zero. Now, we're dividing by really tiny numbers (approaching zero). An integer divided by a very small number, is a very large number. Domain is all real numbers, but cannot equal any multiple of . where n is any integer. Range 0Graph

This is very similar to the regular cos graph, except that its period is changed, or quished." Its usual period of is now .

0Graph

This is similar to the regular sin graph, but it is shifted UP by one unit.

0Graph

is very different from . Radians are used unless otherwise specified. By convention, the variable typically implies use of degrees).
has a period of , meaning that going units in x in either direction will return the same result
etc...
so this is the same graph as

0Graph

The number out front here determines the amplitude. In this case, it's 3. So from minimum to maximum, this graph will cover 6 units total. Multiplying the x here by 2 changes the period from to .

0Graph

The positive 4 there moves the graph over 4 units to the left. The -1 in that location means the graph is shifted down 1. This is the same graph as the previous problem, but shifted down 1 and 4 to the left.

0Simplify

This is equivalent to , which would give an exponent of , or

0Rewrite

0Combine the exponents of

0Simplify

0Solve for x,

0Prove

ADVANCED NOTE: This is the exponential expression of the trig identity

0Given , find

0Solve for x,

0Simplify

0Solve for x,

x = 1 0Solve for y,

0Solve for x,

0Simplify

Here, the base is irrelevant. 0Solve for ,

The second solution, -9, would be outside of the domain of . For , x must be greater than 0. So

0Prove

Let and

0Find the roots of

Two identical roots, 0Find the asymptote(s) of

We already found the roots on the previous problems. The asymptote is where the function is undefined. Here, it's when the denominator is equal to zero. We know that the denominator is equal to zero when . The function goes to positive infinity when approaching the root () from the positive direction. The function also goes to positive infinity when approaching the root from the negative direction (The degree of the polynomial in the denominator is even). 0Graph

We already determined the asymptote behavior, what about the edge behavior? As x gets very large and positve, y approaches zero. As x gets very large and negative, y also approaches zero from the positive side.

0Find the asymptote(s) of

The function is undefined at x = 0. The function goes to positive infinity when approaching zero from the positive direction. The function goes to negative infinity when approaching zero from the negative direction (Remember, the polynomial in the denominator is of odd degree). 0Find the asymptote(s) of

. The function is undefined when . when

There are an infinite number of asymptotes for and those asymptotes are at odd integer multiples of 0Graph

This is similar to the graph of , but it is shifted 5 to the left.

0Graph

Even degree polynomial in the denominator. As x gets very large in either direction, y approaches zero. There are two asymptotes here, -2 and 2. That gives us 4 different cases for asymptotic behavior. Approaching -2 from the left, Approaching -2 from the right, approaching 2 from the left, and approaching 2 from the right. Plugging in values to test for these cases.

0Graph

Asymptotes are at same locations as previous problem, so we can find the asymptote behavior in a similar fashion

0Graph

Remember the properties of . Domain , , and is an increasing function. As x gets larger, f(x) approaches zero and f(x) has an asymptote at x = 1.

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*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Review of Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Intro 0:00
What is a Function 0:10

Domain and Range
Vertical Line Test
Example: Vertical Line Test

Function Examples 1:57

Example: Squared
Example: Natural Log
Example: Exponential
Example: Not Function

Odd and Even Functions 4:39

Example: Even Function
Example: Odd Function

Odd and Even Examples 6:48

Odd Function
Even Function

Increasing and Decreasing Functions 10:15

Example: Increasing
Example: Decreasing

Increasing and Decreasing Examples 11:41

Example: Increasing
Example: Decreasing

Types of Functions 13:32

Polynomials
Powers
Trigonometric
Rational
Exponential
Logarithmic

Lecture Example 1 15:55
Lecture Example 2 17:51
Additional Example 3
Additional Example 4

College Calculus 1 Online Course

Section 1: Overview of Functions
	Review of Functions	26:29
	Compositions of Functions	12:29
Section 2: Limits
	Average and Instantaneous Rates of Change	20:59
	Limit Investigations	22:37
	Algebraic Evaluation of Limits	28:19
	Formal Definition of a Limit	23:39
	Continuity and the Intermediate Value Theorem	19:09
Section 3: Derivatives, part 1
	Limit Definition of the Derivative	22:52
	The Power Rule	26:01
	The Product Rule	14:54
	The Quotient Rule	19:17
	Applications of Rates of Change	17:43
	Trigonometric Derivatives	26:58
	The Chain Rule	23:47
	Inverse Trigonometric Functions	27:05
	Equation of a Tangent Line	15:52
Section 4: Derivatives, part 2
	Implicit Differentiation	30:05
	Higher Derivatives	13:16
	Logarithmic and Exponential Function Derivatives	17:42
	Hyperbolic Trigonometric Function Derivatives	14:30
	Related Rates	29:05
	Linear Approximation	23:52
Section 5: Application of Derivatives
	Absolute Minima and Maxima	18:57
	Mean Value Theorem and Rolle's Theorem	20:00
	First Derivative Test, Second Derivative Test	27:11
	L'Hopital's Rule	23:09
	Curve Sketching	40:16
	Applied Optimization	25:37
	Newton's Method	25:13
Section 6: Integrals
	Approximating Areas and Distances	36:50
	Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus	22:02
	Substitution Method for Integration	23:19
Section 7: Application of Integrals, part 1
	Area Between Curves	19:59
	Volume by Method of Disks and Washers	24:22
	Volume by Method of Cylindrical Shells	30:29
	Average Value of a Function	16:31
Section 8: Extra
	Graphs of f, f', f''	23:58
	Slope Fields for Differential Equations	18:32
	Separable Differential Equations	17:04

Related Books

	Calculus by Larson, 9th Edition Authors: Ron Larson, Bruce H. Edwards ISBN: 0547167024 Publisher: Brooks Cole Year: 2009 The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 1 Author: Ron Larson ISBN: 547213093 Publisher: Brooks Cole Year: 2008 This solutions manual includes worked out solutions to every odd-numbered exercise in Calculus of a Single Variable, 9th edition.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 2 Authors: Ron Larson, Bruce H. Edwards ISBN: 547213107 Publisher: Brooks Cole Year: 2009 This solutions manual includes worked out solutions to every odd-numbered exercise in Multivariable Calculus, 9th edition.

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Course Index

Professor Switkes

Review of Functions

Slide Duration:

Table of Contents

Section 1: Overview of Functions

Review of Functions

26m 29s

Intro