1 / 12

Work on questions for 1-2 minutes

Inc. horizontal tangent at x = -3, x = -0.7 and x = 3. Inc. Dec. Inc. Increasing for x < -3 , -0.7 < x < 3 , x > 3. Decreasing for -3 < x < -0.7. horizontal tangent : f’(x) = 0. Increasing: f’(x) > 0. Decreasing f’(x) < 0. Work on questions for 1-2 minutes.

odin
Download Presentation

Work on questions for 1-2 minutes

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Inc horizontal tangent at x = -3, x = -0.7 and x = 3 Inc Dec Inc Increasing for x < -3 , -0.7 < x < 3 , x > 3 Decreasing for -3 < x < -0.7 horizontal tangent : f’(x) = 0 Increasing: f’(x) > 0 Decreasing f’(x) < 0 Work on questions for 1-2 minutes

  2. 4.1 Increasing and Decreasing FUnctions • Identify properties of a graph including intervals of increase and decrease, turning points, points of inflection and asymptotes. • Identify the intervals of increase and decrease and the turning points of a function. • Use the turning points and intervals of increase and decrease to graph a function. Please read these

  3. the y-values increase which means f’(x) > 0 the y-values decrease which means f’(x) < 0 f’(x) = 0 the function switches from increasing to decreasing or decreasing to increasing. find where f’(x) = 0 and where f’(x) is undefined. Create an interval chart for the derivative using these values. Please copy

  4. Identify the intervals of increase and decrease and the turning points of a function. • Use the turning points and intervals of increase and decrease to graph a function. Example 1: Finding Intervals of Increase/Decrease from an equation (Interval Chart) Determine the intervals of increase and decrease for . Solution is on the next slide On your own: 1) calculate the derivative 2) find when f’(x) = 0 or where f’(x) is undefined 3) create an interval chart (if you remember how to)

  5. Determine the intervals of increase and decrease for . calculate the derivative Therefore is a factor. Synthetic division Finish factoring with decomposition and common factoring Summarize Continued on next slide

  6. Create an interval chart using the derivative and the x values x = 1 and x = -5/4. Intervals Each factorof the derivative If the derivative is positive the function increases If the derivative is negative the function decreases Intervals of increase: Intervals of decrease: and

  7. Identify the intervals of increase and decrease and the turning points of a function. • Use the turning points and intervals of increase and decrease to graph a function. Example 2: Using an interval chart to identify local minimums and maximums Using your interval chart from question 1 , determine the coordinates of all points where f’(x) = 0 and classify these points as local minimums, local maximums or neither.Sketch your results Solution is on the next slide On your own: 1) look at the interval chart and find where the function switched from inc to dec OR dec to inc. 2) decide each time f’(x) = 0 if the point is a max, min or neither. 3) find the y=coordinates of all the point where f’(x) = 0, also find the y-intercept. Sketch the graph.

  8. Using your interval chart from question 1 , determine the coordinates of all points where f’(x) = 0 and classify these points as local minimums, local maximums or neither.Sketch your results From last question, Do not recopy at x = -5/4 the function switched from increasing to decreasing. Since f’(x) = 0 at this point, this point is a local maximum. at x = 1the function does not switch directions. Since f’(x) = 0 at this point, this point is neither a minimum nor a maximum but is just a horizontal tangent location. Substituting the x-values into the original equation the points are The y-intercept is (0,4) Graph on next slide

  9. Identify the intervals of increase and decrease and the turning points of a function. • Use the turning points and intervals of increase and decrease to graph a function.

  10. Identify the intervals of increase and decrease and the turning points of a function. • Use the turning points and intervals of increase and decrease to graph a function. Example 3: More intervals of Increase and Decrease – Rational functions Determine the intervals of increase and decrease for . Determine the coordinates of local minimums and maximums. Solution is on the next slide On your own: 1) calculate the derivative 2) find when f’(x) = 0 or where f’(x) is undefined 3) create an interval chart 4) find the coordinates of the minimums and maximums

  11. Determine the intervals of increase and decrease for . Determine the coordinates of local minimums and maximums. calculate the derivativewith quotient rule Simplify the derivative Summarize the derivative is zero for values of x that make the numerator equal 0 but that do not make the denominator equal to 0 (the derivative is undefined for values x that make the denominator equal to 0.) Continued on next slide

  12. Create an interval chart using the derivative and the x values x = 0, 1 and 2. Intervals of increase: Intervals of decrease: and At x = 0 the function shifts from increasing to decreasing. Since f’(x) = 0 at this point, f(0) = 0 is a local maximum. At x = 1 there is a vertical asymptote. At x = 2 the function shifts from decreasing to increasing. Since f’(x) = 0 at this point, f(2) = 4 is a local minimum.

More Related