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Aim: Functions & Derivative Graphs Course: AP Calculus Do Now: Aim: What is the relationship between the graph of a derivative function and the graph of the original function? Sketch the derivative of the function given by the following graph:

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Aim: Functions & Derivative Graphs Course: AP Calculus Sketching the Derivative derivative is positive when original is increasing. derivative is negative when original is decreasing. pay attention to steep parts. f(x)f(x) derivative is zero when original is at relative maximum/minimum. f(x)f(x)

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs Sketch the graph of this functions derivative

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs Sketch the graph of this functions derivative

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation Sketch the graph of this functions derivative

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs Use the following info to graph f over [-5, 6] 1) f is made of closed line segments end to end 2) graph starts at point (-5, 1) 3) derivative of f is step function shown here

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation Which graph is which? Graph shows distance s, velocity v, and acceleration a as function of time. B = position, A = velocity, C = acceleration

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation Which graph is which? Graph shows distance s, velocity v, and acceleration a as function of time. A = position, C = velocity, B = acceleration

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation Which graph is which? Graph shows distance s, velocity v, and acceleration a as function of time. C = position, B = velocity, A = acceleration

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation When is the body's acceleration equal to zero? Graph shows velocity v as function of time t. 2 < t < 3, 6 < t < 7

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation When is the body standing still? Graph shows distance s as function of time t. 1 < t < 2

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation y = h(x) On what interval(s) is the function h positive? a.-2 < x < -.6 b.(-2, -.6) (1, 1.6) c.(-, -2) d.All x values e.Impossible to tell

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation y = h(x) What are the zeros of g(x)? a.x = -2, -0.6 b.x = -2, -0.6, 1, 1.6 c.x = -1.6, 1.3 d.x = -1.6, 0.1, 1.3 e.Impossible to tell

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation a.h has no zeros b.h is undefined at x = 1 c.h is always negative on its domain d.h has no smallest possible value e.all of the above This is the graph of y = h(x). What can you expect concerning the function h(x)?

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation a.h has no zeros b.h is undefined at x = 1 c.h is always negative on its domain d.h has no smallest possible value e.all of the above This is the graph of y = h(x). What can you expect concerning the function h(x)?

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Aim: Functions & Derivative Graphs Course: AP Calculus Derivative Graphs of Position Equation no relative max or min – no zero slope at x > 0 slope is very (-) and gets less negative (-); slope is never positive from l – r: slope is (-) and gets more (-) as x approaches 0 where f is undefined

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