Graph the function $f ( x )$ over the given interval. Partition the interval into 4 subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum $\sum _ { k = 1 } ^ { 4 } f \left( c _ { k } \right) \Delta x _ { k }$, using the indicated point in the kth subinterval for $\mathrm { c } _ { \mathrm { k } }$.
-$f(x)=x^{2}-3,[0,8], \text { midpoint }$

Question

Quizplus · Accepted Answer

To graph the function, we first need to find the critical points and the concavity. Taking the derivative, we get:

$f'(x)=2x$
$f''(x)=2$

Thus, the function is always increasing and concave up.

We can now use this information to sketch the graph:

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We then partition the interval into 4 subintervals of equal length:

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We use the midpoint of each subinterval to evaluate the function and calculate the area of each rectangle:

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Summing the areas, we get:

$\sum _ { k = 1 } ^ { 4 } f \left( c _ { k } \right) \Delta x _ { k } = f(1)\Delta x + f(3)\Delta x + f(5)\Delta x + f(7)\Delta x = (1^2-3)\Delta x + (3^2-3)\Delta x + (5^2-3)\Delta x + (7^2-3)\Delta x = 116\Delta x$

Note that since the intervals have equal length, $\Delta x$ is the same for each rectangle.

The rectangle heights are therefore:

$f(1)=1^2-3=-2$
$f(3)=3^2-3=6$
$f(5)=5^2-3=22$
$f(7)=7^2-3=46$

Hence, the area is given by:

$A=116\Delta x = 116\cdot\frac{8}{4} = 232$

Therefore, the best choice is C, which shows the correct graph and correctly labeled rectangles with the correct area.

Graph the Function f(x)f ( x )f(x) Over the Given Interval ∑k=14f(ck)Δxk\sum _ { k = 1 } ^ { 4 } f \left( c _ { k } \right) \Delta x _ { k }∑k=14​f(ck​)Δxk​

Graph the Function $f ( x )$ Over the Given Interval $\sum _ { k = 1 } ^ { 4 } f \left( c _ { k } \right) \Delta x _ { k }$