Sketching Curves Using Derivatives

Audience 12th Grade Students

Honors Calculus or AP Calculus Students

Students who wish to learn how to hand draw sketches of curves

Anyone who wishes to review curve sketching.Table of ContentsSketching a Curve Using Derivativest that point.

http://www.nipissingu.ca/calculus/tutorials/curves.html6OrientationFind the Y-interceptFind the X-interceptIdentify the AsymptotesFind the ExtremaIntervals of Increasing or Decreasing ValuesFind Inflection PointsIntervals of Concave up and/or concave down

These are all of the things you will need to find in order to sketch a curve! When youre finished click the quiz button to show what you learned.Im Ready to Take the Quiz!Summary

X-intercept: (0,0)Y-intercept: (0,0)Vertical Asymptote: x=1,-1Horizontal Asymptote: noneOblique Asymptote: y=xFinding ExtremaUsing the first derivative test and a sign chart.

Extrema: These are the local minimums and maximums on the graph.4

http://www.biology.arizona.edu/BioMath/tutorials/Functions/Properties.html25Sorry Go Back and Review the Lesson Again!How can I learn this all by myself!?

Hi! My name is Apple! Ill be helping you through sketching curves today.

http://www.dogcanyon.org/2010/05/14/american-as-apple-pie/7Intervals of Concave Up and/or Concave DownAgain, once you have done the second derivative test, this is an easy one!

Using the sign chart, the intervals with a + sign are concave up, while the intervals with a sign are concave down.7QUIZ TIME!Quick hints!Remember what your basic functions look likeRemember how to do your derivativesMost of this is review, take your time and put it all together!

Use everything you have learned to draw this curve.

Take your time, you can do this!Find the Y-Intercept(1,0)

(0,1)

Undefined

(0,0)

A.B.C.D.

What was your answer?

Your oblique asymptote is not quite right! Let me show you!

Your horizontal asymptote is not quite right! Let me show you!

Undefined!

Therefore there is no horizontal asymptoteVertical Asymptote: x=-2

Your vertical asymptote is not quite right! Let me show you!

Find the x-intercept(2,0)

(0,0)

(2,0) and (-5,0)

(-5,0)A.B.C.D.

What was your answer?HA: noneVA:OA:

HA: noneVA: noneOA:

HA: HA: noneVA: VA:OA: OA:

What was your answer?

D.C.B.A.

First Derivative TestStep 1: Find the First Derivative

Example:

This should be a review of the quotient rule!

First Derivative TestStep 2: Set the numerator of the first derivative equal to 0, and then solve for x.

Example:

First Derivative TestStep 3: Put the x values found back into the original function in order to find the critical points.

Example:

Critical Point: (0,2)

Making a Sign ChartStep 1: Make a number line using all critical numbers

Critical Numbers: These are all the critical values of x, including the x-intercept, any asymptote (with an x value), and values found using the first derivative test. -1 0 1Making a Sign ChartStep 2: Determine which points are actually on the function f(x)

Put all critical values of x into the original equation. If a y value is undetermined then the point is not on the curve. -1 0 1 O ONot on f(x)On f(x)Making a Sign ChartStep 3: Pick a point between the critical values on the number line. Place this x value into the equation f(x), and determine if the y value is + or -. Place your finings on your chart like so: -1 0 1 - + O + O -What are the Extrema?All of the critical values of x that are on the curve f(x) (these are the ones with O), which have a change in sign on the sign chart. This means that the values change from to +, or the values change from + to -.

In this example there is one critical value that : x=1

The extrema is (1,y)! Note: This value of y is from the original equation f(x) -1 0 1 - + O + O -Find the extrema!

Now its your turn to try!

A.B.C.D.

What was your answer?

Help Continued

Help Continued

Help Continued 0 - O + O -

Add the new information to your sign ChartThis is what it should look like:

We can now start to see what the curve is looking like. -1 0 1 - + O + O -The Second Derivative TestThese are the same steps as the first derivative test:

Find the second derivative Set the derivative equal to 0 Find the critical values of x

Make a new sign chart using the same steps as before:

Make a number line with the critical x values Determine if the point is on f(x) Look at a point in the intervals between critical points, and determine if they are positive (+) or negative (-).What Points are Inflection Points?Just like when we found the extrema!

Like before, at the points where the sign changes, these are the x values for the inflection points.

Place these x values into the original f(x) to get the points.What is this concave thing?Its easier to think of concave up as a bowl, and concave down as a hill.

Concave Up Concave Down

*When you think about this, the inflection pointis where the curve changes from concave up,to concave down or vice versa.Are You Ready?

Now you have everything you need to draw the curve. Just put it all together!Im not ready! I need to go back an review sections first.Im ready for that quiz!

http://www.wolframalpha.com/48Summary

+ O - - O - - O +

Summary

-1 0 1 - + O - +