a. If f is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of f(x) dx for any n. Is the statement true or false? Explain. O A. True; the Riemann sum gives an approximation of an integral and never an exact value. O B. False; the left and right Riemann sums only give an exact value for values of n that are very large. OC. False; only a midpoint Riemann sum will give an exact value of the integral. O D. True; the area under a constant function is a rectangle, so the rectangles of a Riemann sum cover exactly the whole area. b b. If f is a linear function on the interval [a,b], then a midpoint Riemann sum gives the exact value of f(x) dx for any n. Is the statement true or false? Explain. O A. True; for each Riemann sum rectangle, the part of the rectangle that is beyond the linear function is equal to the area under the linear function and beyond the rectangle. O B. False; only a left or right Riemann sum gives an exact value of the integral. O C. False; the only time a Riemann sum give an exact value of an integral is when f is a constant function. O D. True; the Riemann sum gives an approximation of an integral and never an exact value. 21/a 2x/a c. Is the equation sin ax dx = cos ax dx = 0 true or false? Explain. O A. False; each integral covers one period of the given trigonometric function, but since a cosine graph is the same as a shifted sine graph by an amount less than one period, the net areas are different. O B. 2x/a 21/a False; sin ax dx # cos ax dx. O C. True; since cosine is the derivative of sine, and the integral of sine will be a constant function, the integral of cosine must be 0. O D. True; each integral covers one period of the given trigonometric function, a cosine graph is the same as a shifted sine graph, which does not affect the net area, and the net areas of both graphs are 0. d. If f(x) dx = |f(x) dx, then f is a constant function. Is the statement true or false? Explain. b b a O A. False; if f(x) dx = f(x) dx, then f(x) dx = f(x) dx = 0. O B. True; if f(x) dx = f(x) dx, then f(x) = - 1. O C. False; f(x) dx never equals f(x) dx. b O D. True; if f(x) dx = f(x) dx, then f(x) = 0. b. b b b e. The property cf(x) dx = c f(x) dx, for any constant c, implies that the equation xf(x) dx =x f(x) dx is true. Is the statement true or false? Explain. A. True; the implication is clear. B. False; the property only allows a constant to be factored out, not a variable. b. OC. False; xf(x) dx = x dx f(x) dx. a O D. True; for each value of x, one can consider x to be constant. Since the statement is true at each each x, it is true for all x.