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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.

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Overview

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  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!!!

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