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 $c _ { k }$.
-$$f ( x ) = x ^ { 2 } - 3 , [ 0,8 ] \text {, left-hand endpoint }$$

Question

Quizplus · Accepted Answer

The graph of $f(x)=x^2-3$ over $[0,8]$ is a parabola opening upwards with vertex at $(0,-3)$. Partitioning the interval into 4 subintervals of equal length, we have $\Delta x = 2$.

Using the left-hand endpoint, we have the following sample points and their corresponding rectangles:

- $k=1, q_1=0, c_1=0, f(c_1)=-3$, rectangle has base $\Delta x$ and height $-3$.
- $k=2, q_2=2, c_2=2, f(c_2)=1$, rectangle has base $\Delta x$ and height $1$.
- $k=3, q_3=4, c_3=4, f(c_3)=13$, rectangle has base $\Delta x$ and height $13$.
- $k=4, q_4=6, c_4=6, f(c_4)=33$, rectangle has base $\Delta x$ and height $33$.

Adding these rectangles to the graph of $f(x)$, we get the picture in choice C.

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 $\sum _ { k = 1 } ^ { 4 } f \left( q _ { k } \right) \Delta x _ { k }$