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 $\sum _ { k = 1 } ^ { 4 } f \left( q _ { k } \right) \Delta x _ { k }$, using the indicated point in the kth subinterval for $c _ { k }$.
-$$f ( x ) = - 4 x ^ { 2 } , [ 0,4 ] \text {, midpoint }$$

Question

Quizplus · Accepted Answer

The graph of $f(x)=-4x^2$ over the interval [0,4] is a downward facing parabola. Partitioning the interval into 4 subintervals of equal length gives us $\Delta x_1=\Delta x_2=\Delta x_3=\Delta x_4=1$.

Using the midpoint rule, we have the following choices for $q_k$ and $c_k$:
\begin{align*}
q_1&=0.5, &c_1&=0, &f(c_1)&=0\
q_2&=1.5, &c_2&=1, &f(c_2)&=-4\
q_3&=2.5, &c_3&=2, &f(c_3)&=-16\
q_4&=3.5, &c_4&=3, &f(c_4)&=-36\
\end{align*}

To evaluate the Riemann sum, we calculate:
$$\sum _ { k = 1 } ^ { 4 } f \left( q _ { k } \right) \Delta x _ { k }=-4(0)(1)-4(-4)(1)-4(-16)(1)-4(-36)(1)=56$$

Among the given choices, only A has the correct placement of rectangles and the correct height of the rectangles, which correspond to the values of $f(c_k)$. Therefore, A is the best choice.

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

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