Applications of Differentiation Curve Sketching

Why do we need this? The analysis of graphs involves looking at “interesting” points and intervals and at horizontal and vertical asymptotes. We use calculus techniques to help us find all of the important aspects of the graph of a function so that we don’t have to plot a large number of points. Curve Sketching

Graphing Skills AKA skills you will have when this chapter has ended  Domain and Range  Symmetry  x and y intercepts  Asymptotes  Extrema  Inflection Points Curve Sketching

4 Extrema on an Interval Extrema of a Function ● Relative Extrema and Critical Numbers ● Finding Extrema on a Closed Interval Curve Sketching

Global Extrema Example Curve Sketching

Lets look at the following function: How many horizontal tangent lines does this curve have? What is the slope of a horizontal tangent? Curve Sketching

Finding Horizontal Tangents  Differentiate:  Set the derivative equal to zero (slope =0):  Solve for x:  This is the x coordinate where you have a horizontal tangent on your graph. Curve Sketching

Finding Horizontal Tangents  Differentiate:  Set the derivative equal to zero (slope =0):  Solve for x:  This is the x coordinate where you have a horizontal tangent on your graph. Curve Sketching

Finding Horizontal Tangents  Differentiate:  Set the derivative equal to zero (slope =0):  Solve for x:  This is the x coordinate where you have a horizontal tangent on your graph. Curve Sketching

Vertical Tangents  Keep in mind where the derivative is defined:  When x = 0 we have slope that is undefined  This is where we have a vertical tangent line.  This counts as a critical number as well and should be considered as such. Curve Sketching

Definition of Extrema Let f be defined on an interval I containing c. 1.f(c) is the maximum of f on I if f(c) > f(x) for all x on I. 2.f(c) is the minimum of f on I if f(c) < f(x) for all x on I. The maximum and minimum of a function on an interval are the extreme values, or extrema, of the function on the interval. The maximum and minimum of a function on an interval are also called the absolute maximum and absolute minimum on the interval, respectfully. Curve Sketching

Definition of a Critical Number Critical numbers are numbers you check to locate any extrema. Let f be defined at c. If f ‘(c) = 0 or if f is undefined at c, then c is a critical number Curve Sketching

Finding Extrema on a Closed Interval  Step 1: find the critical numbers of f in (a,b)  Step 2: Evaluate f at each critical number in (a,b)  Step 3: Evaluate f at each endpoint of [a,b]  Step4: The least of the values is the minimum, the greatest is the maximum. Curve Sketching

Theorem: Relative Extrema Occur only at Critical Numbers If f has a relative minimum or relative maximum at x = c, then c is a critical number of f. These are also known as locations of horizontal tangents Curve Sketching

Critical Numbers  Let f be defined at c. if f’(c)=0 or if f is not differentiable at c, then c is a critical number of f.  Relative extrema (max or min) occur only at a critical number. Curve Sketching

Relative Extrema Curve Sketching

Additional Extrema on a Closed Interval-examples:  Curve Sketching

Locate the Absolute Extrema

19 Finding Extrema on a Closed Interval Left Endpoint Critical Number Right Endpoint f(-1)=7f(0)=0f(1)=-1 Minimum f(2)=16 Maximum Find the extrema: Differentiate the function: Find all values where the derivative is zero or UND: Given endpoints, consider them too!!! Curve Sketching

Homework for 3.1  Page #1-46  #61-64 are physics problems  #65-70 are good thinking problems Curve Sketching

First Derivative Test (Ch.3 S.3)  Finding the critical numbers: C is called a critical number for f if f’(c)=0 (horizontal tangent line) or f’(c) does not exist Curve Sketching

Increasing or Decreasing functions  A function f is called increasing on a given interval (open or closed) if for x 1 any and x 2 in the interval, such that x 2 > x 1, f(x 2 )>f(x 1 )  A function f is called decreasing on a given interval (open or closed) if for x 1 any and x 2 in the interval, such that x 2 > x 1, f(x 2 )<f(x 1 ) Curve Sketching

Maximums, Minimums, Increasing, Decreasing Let the graph represent some function f a bdefc Curve Sketching

Copyright © Houghton Mifflin Company. All rights reserved Definitions of Increasing and Decreasing Functions and Figure 3.15

Copyright © Houghton Mifflin Company. All rights reserved Theorem 3.6 The First Derivative Test

Finding Intervals Where the Function is Increasing/Decreasing Curve Sketching

Finding Intervals Where the Function is Increasing/Decreasing Curve Sketching

Finding Intervals Where the Function is Increasing/Decreasing Curve Sketching

Increasing and Decreasing Example Curve Sketching

LOCAL (RELATIVE) EXTREME VALUES  If x = c is not at an endpoint, then f (c) is a local (relative) maximum value for f(x) if f(c) > or = f(x) for all x-values in a small open interval around x = c.  If x = c is not at an endpoint, then f (c) is a local (relative) minimum value for f(x) if f(c) < or = f(x) for all x-values in a small open interval around x = c. Curve Sketching

Finding local extrema Curve Sketching

Local Extrema Example  See text for additional examples Curve Sketching

Homework for 3.3  pg. 186 #9-38, 43-50, (good for AP) Curve Sketching

3.4 Concavity and the Second Derivative Test Second Derivative Test

Copyright © Houghton Mifflin Company. All rights reserved Definition of Concavity and Figure 3.24

The graph of f is called concave up on a given interval if for any two points on the interval, the graph of f lies below the chord that connects these points. Concavity

The graph of f is called concave down on a given interval if for any two points on the interval, the graph of f lies above the chord that connects these points.

Copyright © Houghton Mifflin Company. All rights reserved Theorem 3.7 Test for Concavity

Copyright © Houghton Mifflin Company. All rights reserved Definition of Point of Inflection and Figure 3.28

Copyright © Houghton Mifflin Company. All rights reserved Theorem 3.8 Points of Inflection

Copyright © Houghton Mifflin Company. All rights reserved Theorem 3.9 Second Derivative Test and Figure 3.31

Second Derivative Test  Finding the points of inflection: C is called a point of interest for f if f’’(c)=0 or f’(c) does not exist.  NOTE: The fact that f’’(x) = 0 alone does not mean that the graph of f has an inflection point at x.  We must find the points where the concavity changes.

Points of Inflection A point of inflection is a point where the concavity of the graph changes from concave up to down or vice-versa.

Concavity and the Second Derivative Test Curve Sketching

Homework for 3.4  P. 195 #1-52, (good for AP prep) Curve Sketching

Limits at Infinity (3.5) Curve Sketching

What happens at x = 1? What happens near x = 1? As x approaches 1, g increases without bound, or g approaches infinity. As x increases without bound, g approaches 0. As x approaches infinity g approaches 0. asymptotes

xx-11/(x-1)^ , , , Undefined , , , vertical asymptotes

xx-11/(x-1)^2 10Undefined , , , ,000, ,000, ,000, horizontal asymptotes

AKA Limits at infinity  We know that  And we know that  So we can expand on this… horizontal asymptotes

Theorem:  If r is a positive rational number and c is any real number, then  Furthermore, if is defined when x<0, then horizontal asymptotes

Theorem:  The line y=L is a horizontal asymptote of the graph of f if or Limits at Infinity

Examples:  Finding a limit at infinity Limits at Infinity

Finding a limit at infinity  Direct substitution yields indeterminate form!  Now we need to do algebra to evaluate the limit:  Divide each term by x:  Simplify:  Evaluate (keep in mind each 1/x  0) Limits at Infinity

A comparison (AKA the shortcut!)  Direct substitution yields infinity/infinity!  Do algebra: Notice a pattern????? Limits at Infinity

A comparison (AKA the shortcut!)  Direct substitution yields infinity/infinity!  Do algebra: Bottom HeavyBalanced Top Heavy Limits at Infinity

Limits with 2 horizontal asymptotes

Homework  Page 205 #2-38E Curve Sketching

Curve Sketching 3.6  The following slides combine all of our new graphing and analysis skills with our precalculus skills

Example: Curve Sketching

Asymptotes: Polynomial functions do not have asymptotes. a) Vertical: No vertical asymptotes because f(x) is continuous for all x. b) Horizontal: No horizontal asymptotes because f(x) is unbounded as x goes to positive or negative infinity. Curve Sketching

Intercepts: a) y-intercepts: f(0)=1 y-intercept: (0,1) b) x-intercepts: difficult to find – use TI-89 Curve Sketching

Critical numbers: a) Take the first derivative: b) Set it equal to zero: c) Solve for x: x=1, x=3 Curve Sketching

Critical Points:  a)Critical numbers x=1 and x=3  b) Find corresponding values of y:  c) Critical points: (1,5) and (3,1) Curve Sketching

Increasing/Decreasing: a) Take the first derivative: b) Set it equal to zero: c) Solve for x: x=1, x=3 Curve Sketching

d) Where is f(x)undefined? Nowhere e) Sign analysis: Plot the numbers found above on a number line. Choose test values for each interval created and evaluate the first derivative: Increasing/Decreasing:

Extrema f’(x) changes from positive to negative at x=1 and from negative to positive at x=3 so and are local extrema of f(x). Note: Values of corresponding to local extrema of must:  Be critical values of the first derivative – values at which equals zero or is undefined,  Lie in the domain of the function, and  Be values at which the sign of the first derivative changes. Curve Sketching

 f(x) is increasing before x=1 and decreasing after x=1. So (1,5) is a maximum.  f(x) is decreasing before x=3 and increasing after x=3. So (3,1) is a minimum. First Derivative Test Curve Sketching

SECOND DERIVATIVE TEST Alternate method to find relative max/min: a) Take the second derivative: b) Substitute x-coordinates of extrema: (negative local max) (positive local min) c) Label your point(s): local max: (1,5) local min: (3,1) Curve Sketching

Concave Up/Down Curve Sketching

Inflection Points Curve Sketching

Create a table of the values obtained: Curve Sketching x0123 f(x) f’(x) f’’(x)

Curve Sketching

Graphing Functions

Domain and Range  Domain : evaluate where the function is defined and undefined, values of discontinuity (under radicals, holes, etc.), nondifferentiability (sharp turns, vertical tangent lines and points of discontinuity).  Range : evaluate the values of y after graphing. Look for minimum and maximum values and discontinuities vertically.

Domain and Range 

Finding the Range of a Function We will look at this AFTER we graph the equation using calculus, you can graph it on the calculator now if you like. Curve Sketching

Symmetry About the y-axis: Replace x with –x and you get the same equation after you work the algebra About the x-axis: Replace y with –y and you get the same equation after you work the algebra About the origin: Replace x with –x and y with –y and get the same equation after you work the algebra Curve Sketching

Graphing Functions Do we have symmetry about the y-axis? replace x with –x and we get: Are these equivalent equations? NO! So we do not have symmetry about the y-axis. Curve Sketching

Graphing Functions Do we have symmetry about the x-axis? replace y with –y and we get: Are these equivalent equations? NO! So we do not have symmetry about the x-axis. Curve Sketching

Graphing Functions Do we have symmetry about the origin? replace x with –x and y with –y and we get: Are these equivalent equations? NO! So we do not have symmetry about the origin. Curve Sketching

x and y-intercepts  x-intercepts: Set top of the fraction equal to 0 in the original equation and solve for x values.  y-intercepts: Substitute 0 in for x and solve for y values.

x-intercepts Replace y with 0 and we get ??? so do we cross the x- axis?

y-intercepts Replace x with 0 and we get y = -2 so we cross the y-axis at (0,-2) Curve Sketching

Horizontal Asymptotes The line y=L is called a horizontal asymptote for f if the limit as x approaches infinity from both the left and/or right is equal to L In other words, find the original equation as x approaches positive or negative infinity. Limits at Infinity Curve Sketching

Horizontal Asymptotes As x goes to infinity what happens to our equation? It goes to infinity as well. So we have no horizontal asymptotes Curve Sketching

Vertical Asymptotes The line x=c is called a vertical asymptote for f at x=c if the limit goes to positive or negative infinity as x approaches c from the positive and/or negative side. This is the complicated way of saying set the denominator (if there is one) to zero and find the x values. Infinite Limits Curve Sketching

Vertical Asymptotes We set the denominator to equal zero and find that we have an asymptote at x = 2 Curve Sketching

Slant Asymptotes Given a function that consists of a composition of functions: If there are no common factors in the numerator and denominator, perform long division. The NONfractional part gives you the equation of the slant asymptotes. Curve Sketching

Graphing Functions Performing long division: we get a non- fractional part of the answer of y = x for our slanted asymptote Curve Sketching

First Derivative Test Take the derivative of the function, set it equal to 0 and find the corresponding values of x. These are our CRITICAL POINTS (Potential extrema)

Graphing Functions Take the second derivative Set it equal to 0 Solve for x This is our potential point of inflection

Graphing Functions Here we compile all of the information to enable us to give a pretty accurate graph WITHOUT technology

Organize the Information xf(x)f’(x)f’’(x)Conclusion -∞<x<0+-Increasing, concave down X=0-20-Relative Max 0<x<2--Decreasing, concave down X=2Undef Vertical asymptote 2<x<4-+Decreasing, concave up X=460+Relative min 4<x<∞++Increasing, concave up

Homework:  Page 215 #1-45,  (67-71 are good thinking problems) Curve Sketching