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Before watching the videos, think about and answer these questions to the best of your ability. Note: in these questions, the left hand side of the first interval is denoted by $x_0$.
Consider the function $f(x)=3x$. Which of the following is a correct expansion of the Riemann sum $\sum f(x) \Delta x$ over the interval $[6,8]$ using four subdivisions of equal length?
$3 \times 6 \times \frac {1}{2} + 3 \times 6.5 \times \frac {1}{2} + 3 \times 7 \times \frac {1}{2} + 3 \times 7.5 \times \frac {1}{2}$ $3 \times 6.5 \times \frac {1}{2} + 3 \times 7 \times \frac {1}{2} + 3 \times 7.5 \times \frac {1}{2} + 3 \times 8 \times \frac {1}{2}$ $3 \times 6 \times \frac {1}{2} + 3 \times 6.5 \times \frac {1}{2} + 3 \times 7 \times \frac {1}{2} + 3 \times 7.5 \times \frac {1}{2} + 3 \times 8 \times \frac {1}{2}$ Both a. and b. are correct. a., b. and c. are all correct. At least one of a., b. and c. is correct IF you remove the repeating $\frac {1}{2}$ None of the above
Which notation best represents the left Riemann sum for the function $r(t) = 4.2 e^{0.5t}$ over the interval $[2,4]$ using four subdivisions of equal length $\Delta t$ where $t_0 = 2$, $t_1 = 2.5$, and so on?
$\sum _{k=1}^4 4.2 e^{0.5 t_k} \Delta t$ $\sum _{k=0}^4 4.2 e^{0.5 t_k} \Delta t$ $\sum _{k=0}^3 4.2 e^{0.5 t_k} \Delta t$ $\sum _{k=2}^4 4.2 e^{0.5 t_k} \Delta t$ None of the above
Suppose that $f(x)=x$. What is the value of $\sum _{k=1}^3 f(x_k) \Delta x$ over the interval $[1,7]$ using three subdivisions of equal length $\Delta x$ where $x_0=1$ and so on?
18 30 32 96 126 Both a. and b. are correct. a., b. and c. are all correct. At least one of a., b. and c. is correct IF you remove the repeating $\frac {1}{2}$ None of the above