1 / 14

Overview

Overview. Chapter Four. Increasing and Decreasing. This is slope so you use first derivative. If it asks for intervals of increasing and decreasing f’(x)=0 to find stationary points and also find critical #s. Then make an interval table and test in the first derivative. Concavity.

diella
Download Presentation

Overview

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. Overview Chapter Four

  2. Increasing and Decreasing • This is slope so you use first derivative. • If it asks for intervals of increasing and decreasing f’(x)=0 to find stationary points and also find critical #s. • Then make an interval table and test in the first derivative.

  3. Concavity • This is the second derivative. • If it asks for intervals of concave up and concave down, use f “(x)=0 to solve for possible inflection points. • Then make an interval table and test in the second derivative.

  4. Inflection Point • To find inflection points, set f “(x)=0 to solve for possible inflection points. • Then make an interval table and test in the second derivative. • Where f”(x)=0 AND concavity changes, then that x value is an inflection point.

  5. Relative Extrema • Relative extrema are relative maximums (top of any mountain on a local level) and relative minimums (bottom of any valley on a local level). • These occur at critical numbers. • Critical numbers include cusps, locations of vertical tangent lines, stationary points, etc. • Stationary points occur where the slope of the tangent line is horizontal (f’(x)=0) and changes from increasing to decreasing or vice versa. • Critical numbers occur where f’(x) is undefined.

  6. Are they Minimums or Maximums? • First derivative test (test in first derivative using an interval table) • If f’(x) on your interval table goes from increasing to decreasing, then you have a relative maximum. • If f’(x) on your interval table goes from decreasing to increasing, then you have a relative minimum. • Second derivative test (take the critical numbers, including stationary points) and substitute them into the second derivative (NO INTERVAL TABLE) • If the result of your substitution is a positive number, then f(x) is concave up at that value of x which means you have a relative minimum (local minimum). • If the result of your substitution is a negative number, then f(x) is concave down at that value of x which means you have a relative maximum (local maximum).

  7. Absolute Maximums and Minimums • Absolute maximums and minimums are the overall highest peak in the mountain range and the overall lowest valley. • Therefore, we need to know position. We find position by substituting the following numbers into f(x). • Take the critical numbers, including stationary points from the first derivative and substitute them into the original function. • You must also test endpoints (if there are any) and end behavior if there is not a given interval [a,b].

  8. Optimization • Step 1: Draw and label • Step 2: Formula to maximize or minimize • Step 3: Write that formula as a function of one variable • Step 4: Interval of possible values • Step 5: Find the relative extrema • Take the derivative, set=0, solve for x, determine whether the results are maximum(s) or minimum(s).

  9. Rectilinear Motion • Velocity is the derivative of position • v(t) = s’ (t) = ds/dt • Acceleration is the derivative of velocity • a(t) = v’ (t) = dv/dt • Speed is the absolute value of velocity • speed =

  10. Speeding Up vs. Slowing Down • speeding up when its speed is increasing • slowing down when its speed is decreasing • SIGNS of velocity and acceleration: • same sign speedingup • different sign slowing down • Make a timeline with velocity and acceleration and compare their signs as we did in the example on the next slide

  11. Speeding Up/Slowing Down Analysis Example

  12. Mean Value Theorem • Plug end points into original function to find the y values of your points • Plug two points into slope formula y2-y1/x2-x1 to get the slope of your secant line. • Take the derivative of f(x) to get the slope of your tangent line. • Parallel lines have equal slopes, so set the slope of your secant line=slope of your tangent line: y2-y1/x2-x1 = f’(x) and solve for x.

  13. Rolle’s Theorem • Rolle’s Theorem is a special case of the Mean Value Theorem, so you can follow the same steps. • Plug end points into original function to find the y values of your points • Plug two points into slope formula y2-y1/x2-x1 to get the slope of your secant line which for Rolle’s Theorem will always = 0. • Take the derivative of f(x) to get the slope of your tangent line. • Parallel lines have equal slopes, so set the slope of your secant line which for Rolle’s Theorem will always = 0 =slope of your tangent line: y2-y1/x2-x1 = 0= f’(x) and solve for x.

  14. Good luck on test!!!

More Related