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Unit 11 – Derivative Graphs Section 11.1 – First Derivative Graphs

Unit 11 – Derivative Graphs Section 11.1 – First Derivative Graphs. First Derivative Slope of the Tangent Line. Given the graph of FUNCTION f(x):. Slope of tangent line positive. This is the graph of f(x). This is the graph of f(x). Given the graph of FUNCTION f(x):.

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Unit 11 – Derivative Graphs Section 11.1 – First Derivative Graphs

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  1. Unit 11 – Derivative GraphsSection 11.1 – First Derivative Graphs First Derivative Slope of the Tangent Line

  2. Given the graph of FUNCTION f(x): Slope of tangent line positive This is the graph of f(x) This is the graph of f(x)

  3. Given the graph of FUNCTION f(x): Slope of tangent line negative This is the graph of f(x) This is the graph of f(x)

  4. Given the function If f ‘ (x) is positive: • Slopes of f(x) are positive • f(x) is increasing If f ‘ (x) is negative: • Slopes of f(x) are negative • f(x) is decreasing This is the graph of

  5. This is the graph of f(x) This is the graph of f ‘ (x)

  6. At x = 3….. The graph of f(x) is increasing. At x = -1….. The graph of f(x) is decreasing. This is the graph of f ‘ (x)

  7. On what intervals is the graph of f(x) increasing? X On what intervals is the graph of f(x) decreasing? X X X X X For what values of x is f ‘ (x) = 0? -2 , 0 , 1 , 2 X BONUS QUESTION: This is the graph of f ‘ (x) For what values of x is f “ (x) = 0? -1.2 , 0.4 , 1.5

  8. The First Derivative Test For Maximum/Minimum The solutions to f ‘ (x) = 0 are CRITICAL POINTS. If f ‘ (x) changes from positive to negative, a RELATIVE MAXIMUM exists. If f ‘ (x) changes from negative to postive, a RELATIVE MINIMUM exists.

  9. For what values of x is f ‘ (x) = 0? -2 , 0 , 2 , 1 The critical points of f(x) are -2, 0, 1, 2 The relative maxima of f(x) are at -2 and 1 because f ‘ (x) changes from positive to negative The relative minima of f(x) are at 0 and 2 because f ‘ (x) changes from negative to positive This is the graph of f ‘ (x)

  10. Copy the graph -> X If the graph represents f(x), mark with an x the critical numbers X X X X X If the graph represents f ‘ (x), mark with an x the critical numbers X If the graph represents f(x), estimate to one decimal place the value(s) of x at which there is a relative maximum. -1.4, 0.4 If the graph represents f ‘ (x), estimate to one decimal place the value(s) of x at which there is a relative minimum. -1.9, 1.8

  11. CALCULATOR REQUIRED a) For what value(s) of x will there be a horizontal tangent on f(x) ? 1 b) For what value(s) of x will the graph of f(x) be increasing? c) For what value(s) of x will there be a relative minimum on f(x)? 1 d) For what value(s) of x will there be a relative maximum on f(x)? none

  12. If the graph represented is f(x), for what values of x is the first derivative equal to zero? If the graph represented if f ‘ (x), for what values of x would the local max(s) and local min(s) be? If the graph represented is f(x), write using interval notation the interval(s) on which the graph is increasing. If the graph represented is f ‘ (x), write using interval notation the interval(s) on which the graph is decreasing. -1 and 2 -2, 1, 3 (-3, -1), (2, 4) [-3, -2) U (1, 3] The graph is on the interval [-3, 4]

  13. If the graph represented is f(x), for what value(s) of x if f ‘ (x) = 0? If the graph represented is f ‘ (x), for what values of x is there a relative minimum? If the graph represented is f(x), write using interval notation the interval(s) on which f ‘ (x) is positive. If the graph represented if f ‘ (x), what at value(s) of x is there a relative maximum? -1.5, -0.5, 0.5, 1.5 0 (-2, -1.5), (-0.5, 0.5), (1.5, 2) -1, 1 The graph is on the interval [-2, 2]

  14. Where are the critical point(s) of f(x)? x = 1 What is f ‘ (1)? 0 Where is the ABSOLUTE maximum of f(x) on [-5, 3]? x = 3 This is the graph of f ‘ (x) on the interval [-5, 3] Where is the ABSOLUTE minimum of f(x) on [-5, 3]? x = -5

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