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8.1-8.3 Review: Functions and Max/Min Problems

This review covers various topics in AP Calculus, including analyzing functions, critical values, max/min values, concavity, and asymptotes. It also provides guidance on finding derivatives, drawing number lines, and solving max/min problems.

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8.1-8.3 Review: Functions and Max/Min Problems

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  1. 8.1-8.3 Review: Functions and Max/Min Problems AP Calculus

  2. Analyzing Functions Critical Values: x coordinates of points at which derivative of f is 0 or undefined f(x) reaches relative max/min values when derivative is 0 or undefined (horizontal tangent/cusp) *** f ‘(x) must change sign for rel max/min Changes in concavity may occur when the second derivative f ’’(x) is 0 or undefined. Function is concave up when f ’’ is > 0 Concave down when f ’’ < 0 The point of inflection occurs where the graph changes concavity.

  3. Analyzing Functions Max/Min VALUE of a function: Y value of function. Absolute min/max: Highest or lowest value of function on an interval. Can take place where the derivative is undefined or 0, OR AT INTERVAL ENDPOINTS!!!

  4. Second Derivative Test At a point x, if f ‘(x) = 0 (possible rel. min or max – critical point) and f “(x) < 0 (concave down), f reaches a relative MAXIMUM at x. If f ‘(x) = 0 and f “(x) > 0 (concave up), f reaches a relative MINIMUM at x.

  5. VERTICAL ASYMPTOTES • Vertical Asymptotes: Occur when denominator of function equals 0. Typically can factor or use the quadratic formula to determine. • Vert. Asymptotes: x = -1/2, x = 4

  6. Horizontal Asymptotes Horizontal Asymptotes: Value y approaches as x approaches infinity. So horizontal asymptote occurs at y = 5/3 = 0, so asymp. is y = 0.

  7. Know how to: Find derivatives of functions such as and factor the result to find solutions when f ‘(x) = 0. Draw number lines illustrating f ‘(x) and f “(x) (to show intervals where graphs increase/decrease or are concave up/down. Use chart to identify graph features such as rel. min/max and points of inflection. Draw a sketch of f(x) given f ‘(x) Sketch f(x) given number lines for f ‘(x) and f “(x)

  8. MAX/MIN PROBLEMS Write equation of function to maximize or minimize. Typical examples are area, volume, distance, Pythagorean Theorem Be aware of any limitations. Often, a restriction function allows original function to be re-written using one variable. Make sure function is written using one variable – max/min values occur when f ‘(x) = 0 (or possibly at interval endpoints). Be careful! Draw/label diagrams!!!

  9. Drawing f(x) given f ‘(x)

  10. ANSWER

  11. Drawing f(x) given f ‘(x)

  12. ANSWER

  13. Drawing f(x) given f ‘(x)

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