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4.4: Derivatives and the Shape of a Graph

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    1. Given function \(f(x)\) graphed below, construct sign charts for \(f\), \(f'\), and \(f''\).
      The graph of a smooth, continuous function with zeros at x=-1 and x=3, a critical point at x=1, and no inflection points. For x<-1, the function is positive, decreasing, and concave up.
       
    2. Given function \(f(x)\) graphed below, construct sign charts for \(f\), \(f'\), and \(f''\).
      The graph of a smooth, continuous function with x-intercepts at x=-2 and x=1, critical points at x=-1 and x=1, and an inflection point at x=0. For x<-2, the graph of the function is negative, increasing, and concave down.
       
    3. Given function \(f(x)\) graphed below, construct sign charts for \(f\), \(f'\), and \(f''\).
      The graph of a smooth, continuous function with x-intercepts at x=-1 and x=0, critical points at x=-1 and x=1, and inflection points at x=0 and x=1. For x<-2, the function is positive, decreasing, and concave up.
       
    4. Sketch the graph of a smooth function \(f(x)\) that agrees with the sign charts below.
      f(x) is positive for x<4 and negative for x>4; f'(x) is negative for x<-1, positive for -1<x<3, and negative for x>3; and f''(x) is positive for x<1 and negative for x>1.
       

    5. Sketch the graph of a smooth function \(f(x)\) that agrees with the sign charts below.
      f(x) is negative for x<-4, positive for -4<x<-2, and negative for x>-2; f'(x) is positive for x<-3, negative for -3<x<-1, positive for -1<x<1, and negative for x>1; and f''(x) is negative for x<-2, positive for -2<x<0, and negative for x>0.

    6. Sketch the graph of a smooth function \(f(x)\) that agrees with the sign charts below.
      f(x) is negative for x<-2, positive for -2<x<2, and negative for x>2; f'(x) is positive for x<-2, positive for -2<x<0, negative for 0<x<2, and negative for x>2; and f''(x) is negative for x<-2, positive for -2<x<-1, negative for -1<x<1, positive for 1<x<2, and negative for x>2.
       
    7. Consider the function \(f(x) = x^2 - 2x\). Construct sign charts for \(f\), \(f'\), and \(f''\), then use your sign charts to sketch the graph of \(f\). Be sure to include all \(x\)-intercepts, relative extrema, and inflection points of \(f\) in your graph, and make sure the behavior of your graphed function agrees with your sign charts. Confirm your result using a graphing calculator.
       
    8. Consider the function \(f(x) = x^2 + x + 1\). Construct sign charts for \(f\), \(f'\), and \(f''\), then use your sign charts to sketch the graph of \(f\). Be sure to include all \(x\)-intercepts, relative extrema, and inflection points of \(f\) in your graph, and make sure the behavior of your graphed function agrees with your sign charts. Confirm your result using a graphing calculator.
       
    9. Consider the function \(f(x) = 3 + 2x - x^2\). Construct sign charts for \(f\), \(f'\), and \(f''\), then use your sign charts to sketch the graph of \(f\). Be sure to include all \(x\)-intercepts, relative extrema, and inflection points of \(f\) in your graph, and make sure the behavior of your graphed function agrees with your sign charts. Confirm your result using a graphing calculator.
       
    10. Consider the function \(f(x) = x^3 - 3x\). Construct sign charts for \(f\), \(f'\), and \(f''\), then use your sign charts to sketch the graph of \(f\). Be sure to include all \(x\)-intercepts, relative extrema, and inflection points of \(f\) in your graph, and make sure the behavior of your graphed function agrees with your sign charts. Confirm your result using a graphing calculator.
       
    11. Consider the function \(f(x) = \dfrac{1}{9}(x^4 + 4x^3)\). Construct sign charts for \(f\), \(f'\), and \(f''\), then use your sign charts to sketch the graph of \(f\). Be sure to include all \(x\)-intercepts, relative extrema, and inflection points of \(f\) in your graph, and make sure the behavior of your graphed function agrees with your sign charts. Confirm your result using a graphing calculator.
       
    12. Consider the function \(f(x) = x^4 - x\). Construct sign charts for \(f\), \(f'\), and \(f''\), then use your sign charts to sketch the graph of \(f\). Be sure to include all \(x\)-intercepts, relative extrema, and inflection points of \(f\) in your graph, and make sure the behavior of your graphed function agrees with your sign charts. Confirm your result using a graphing calculator.
       
    13. Consider the function \(f(x) = x^4 - 4x^2 + 3\). Construct sign charts for \(f\), \(f'\), and \(f''\), then use your sign charts to sketch the graph of \(f\). Be sure to include all \(x\)-intercepts, relative extrema, and inflection points of \(f\) in your graph, and make sure the behavior of your graphed function agrees with your sign charts. Confirm your result using a graphing calculator.
       
    14. Consider the function \(f(x) = (x-2)^2(x-4)^2\). Construct sign charts for \(f\), \(f'\), and \(f''\), then use your sign charts to sketch the graph of \(f\). Be sure to include all \(x\)-intercepts, relative extrema, and inflection points of \(f\) in your graph, and make sure the behavior of your graphed function agrees with your sign charts. Confirm your result using a graphing calculator.

    4.4: Derivatives and the Shape of a Graph is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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