Video Set Introduction

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Before watching the videos, think about and answer these questions to the best of your ability.

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?

$\sum f(x) \Delta x = 3 \times 6 \times \frac {1}{2} + 3 \times 6.5 \times \frac {1}{2} + 3 \times 7 \times \frac {1}{2} + 3 \times 7.5 \times \frac {1}{2}$ $\sum f(x) \Delta x = 3 \times 6.5 \times \frac {1}{2} + 3 \times 7 \times \frac {1}{2} + 3 \times 7.5 \times \frac {1}{2} + 3 \times 8 \times \frac {1}{2}$ $\sum f(x) \Delta x = 3 \times 6 \times \frac {1}{2} + 3 \times 6.5 \times \frac {1}{2} + 3 \times 7 \times \frac {1}{2} + 3 \times 7.5 \times \frac {1}{2} + 3 \times 8 \times \frac {1}{2}$ 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 are correct.

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?

$\sum _{k=1}^4 = 4.2 e^{0.5 t_k} \Delta t$ $\sum _{k=0}^4 = 4.2 e^{0.5 t_k} \Delta t$ $\sum _{k=0}^3 = 4.2 e^{0.5 t_k} \Delta t$ $\sum _{k=2}^4 = 4.2 e^{0.5 t_k} \Delta t$

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?

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