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.
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.
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54144759-0e64-4a52-9eb8-f3918d0f39d7%2F86589b8d-2f3f-4ea7-84db-19afdb66980c%2Flclp5o6.jpeg&w=3840&q=75)
Transcribed Image Text: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.
Transcribed Image Text: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.
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