Question 1

Suppose you want to approximate the definite integral $$\$$int_0$$^3 \left(2 - \frac{2}{3}x\right) dx$$ using a left-endpoint Riemann sum with 3 rectangles of equal width. What is the width $$\Delta x of $$each rectangle?

Accepted Answer

$$\Delta x = 1$$

Question 2

Given the function $$f(x) = 2 - \frac{2}{3}x on$$ $$[0,3]$$, which of the following best describes the left-endpoint Riemann sum approximation for $$\$$int_0$$^3 f(x) dx$$ using 3 rectangles?

Accepted Answer

$$f(0)\Delta x + f(1)\Delta x + f(2)\Delta x$$

Question 3

If you use a left-endpoint Riemann sum to approximate $$\$$int_0$$^3 \left(2 - \frac{2}{3}x\right) dx$$, will your estimate be an overestimate or an underestimate of the true value of the integral?

Accepted Answer

It will be an underestimate because the function is decreasing.

Question 4

Which formula gives the right endpoint of the $$ith $$subinterval when dividing $$[a,b]$$ into $$n$$ equal subintervals?

Accepted Answer

$$a + i\Delta x$$

Question 5

Given $$\Delta x = \frac{b-a}{n}$$, what is the height of the $$ith $$rectangle in a right-endpoint Riemann sum for $$f(x) on$$ $$[a,b]$$?

Accepted Answer

$$f(a + i\Delta x)$$

Question 6

What is the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

If $$F(x) = \int_a^x f(t) dt$$, then $$F'(x) = f(x).$$

Question 7

Mirah walks along a straight road. Her velocity $$v(t) is $$given by the graph below. What does the area under the velocity curve between $$t=0$$ and $$t=2$$ represent?

Accepted Answer

Her total displacement from the starting point after 2 hours.

Question 8

If $$F(x) = \$$int_0$$^x \sin(t^2$) dt$$, what is $$F'(x)$$?

Accepted Answer

$$\sin(x^2$)$$

Question 9

Evaluate $$\$$int_0$$^1 x^2$ dx.$$

Accepted Answer

$$\frac{1}{3}$$

Question 10

Which of the following is a correct interpretation of $$\int_a^b f(x) dx in a $$real-world context?

Accepted Answer

The net accumulation of the quantity represented by $$f(x)$$ from $$x=a to$$ $$x=b.$$

Question 11

Suppose $$G(x) = \$$int_1$$^x \sin(t^2$) dt. $$What is $$G'(x)$$?

Accepted Answer

$$\sin(x^2$)$$

Question 12

If $$H(x) = \$$int_0$$^{x^2$} \sin(t^2$) dt$$, what is $$H'(x)$$?

Accepted Answer

$$2x\sin(x^4$)$$

Question 13

Evaluate $$\$$int_1$$^2 \frac{1}{x} dx.$$

Accepted Answer

$$\ln(2)$$

Question 14

Which of the following statements about Riemann sums is true?

Accepted Answer

As the number of rectangles increases, the Riemann sum approaches the exact value of the definite integral.

Question 15

If $$f(x) is $$continuous on $$[a,b]$$, which of the following is guaranteed by the Fundamental Theorem of Calculus?

Accepted Answer

There exists an antiderivative $$F(x)$$ such that $$F'(x) = f(x).$$

Question 16

Mirah's distance from her starting point at time $$t is$$ $$D(t) = \$$int_0$$^t v(s) ds. On $$what interval is $$D(t)$$ increasing?

Accepted Answer

Where $$v(t) \u003e 0$$

Question 17

Evaluate $$\$$int_0$$^1 e^x dx.$$

Accepted Answer

$$e - 1$$

Question 18

Which of the following is the correct formula for the area of the $$ith $$rectangle in a right-endpoint Riemann sum for $$f(x) on$$ $$[a,b]$$?

Accepted Answer

$$f(a + i\Delta x) \cdot \Delta x$$

Question 19

Which of the following is a valid use of geometry to compute $$\$$int_0$$^3 \left(2 - \frac{2}{3}x\right) dx$$?

Accepted Answer

Find the area of the trapezoid under the line from $$x=0 to$$ $$x=3.$$

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