4.4: Derivatives and the Shape of a Graph

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Mar 6, 2024
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- 4.3: The Mean Value Theorem
- 4.5: Limits at Infinity and Asymptotes

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

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