Question 4

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The graph of a differentiable function $f$ is shown for $-6 \le x \le 7$ . It has a horizontal tangent at $x = -2$ and is a straight line for $0 \le x \le 7$ . $R$ is the region in the second quadrant between the graph of $f$ , the line $x = -6$ and the two axes. The area of $R$ is $12$ .

$\text{(a)}$
Let $g(x) = \displaystyle\int_0^x f(t)\,dt$ . Find $g(-6)$ , $g(4)$ and $g(6)$ .
$\text{(b)}$
For $g$ as in part (a), find every $x$ in $0 \le x \le 6$ where the graph of $g$ has a critical point. Give a reason.
$\text{(c)}$
Let $h(x) = \displaystyle\int_{-6}^{x} f'(t)\,dt$ . Find $h(6)$ , $h'(6)$ and $h''(6)$ . Show the working.

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This question is about a function defined by the area under another function's graph

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Question 4

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$\text{(a)}$
Let $g(x) = \displaystyle\int_0^x f(t)\,dt$ . Find $g(-6)$ , $g(4)$ and $g(6)$ .
$\text{(b)}$
For $g$ as in part (a), find every $x$ in $0 \le x \le 6$ where the graph of $g$ has a critical point. Give a reason.
$\text{(c)}$
Let $h(x) = \displaystyle\int_{-6}^{x} f'(t)\,dt$ . Find $h(6)$ , $h'(6)$ and $h''(6)$ . Show the working.

Now playing

Understanding the Question

This question is about a function defined by the area under another function's graph

Part 1 of 4