Calculus Quiz

Let f be the function given by f(x) = 4^x. If four subintervals of equal length are used, what is the value of the left Riemann sum approximation for ∫₁³ f(x) dx?

• 30

• 60

• 62

• 120

The graph of the function f is shown above for 0 ≤ x ≤ 3. Of the following, which has the least value?

• ∫₀³ f(x) dx

• Left Riemann sum approximation of ∫₀³ f(x) dx with 4 subintervals of equal length

• Right Riemann sum approximation of ∫₀³ f(x) dx with 4 subintervals of equal length

• Midpoint Riemann sum approximation of ∫₀³ f(x) dx with 4 subintervals of equal length

• Trapezoidal sum approximation of ∫₀³ f(x) dx with 4 subintervals of equal length

If three equal subdivisions of [-4, 2] are used, what is the trapezoidal approximation of the integral from -4 to 2 of (x^2 * e^(-x/2)) dx?

• e^2 + e^0 + e^-2

• e^4 + e^2 + e^0

• e^4 + 2e^2 + 2e^0 + e^-2

• 1/2 * (e^4 + e^2 + e^0 + e^-2)

• 1/2 * (e^4 + 2e^2 + 2e^0 + e^-2)

The function f is continuous on the closed interval [2,8] and has values that are given in the table above. Using the subintervals [2,5], [5,7], and [7,8], what is the trapezoidal approximation of the integral from 2 to 8 of f(x) dx?

• 110

• 130

• 160

• 190

• 210

Which of the following limits is equal to the integral from 3 to 5 of x^4 dx?

• a

• b

• c

• d

• e

The graph of f', the derivative of the function f, is shown above. If f(0) = 0, which of the following must be true?

I. f (0) > f (1)

II. f (2) > f (1)

III. f (1) > f (3)

• A) I only

• B) II only

• C) III only

• D) I and II only

• E) II only III only

If a trapezoidal sum overapproximates ∫₀⁴ f(x) dx, and a right Riemann sum underapproximates ∫₀⁴ f(x) dx, which of the following could be the graph of y = f(x)?

• A) Graph A

• B) Graph B

• C) Graph C

• D) Graph D

• E) Graph E

The data for the acceleration ( a(t) ) of a car from 0 to 6 seconds are given in the table above. If the velocity at ( t = 0 ) is 11 feet per second, the approximate value of the velocity at ( t = 6 ), computed using a left-hand Riemann sum with three subintervals of equal length, is

• 26 ft/sec

• 30 ft/sec

• 7 ft/sec

• 39 ft/sec

• 41 ft/sec

The table above gives selected values for a continuous function ( f ). If ( f ) is increasing over the closed interval [0,2], which of the following could be the value of ( ∫_0^2 f(x) dx )?

• 36

• 41

• 50

• 53

A tank contains 35 gallons of water at time t = 2 minutes. Water is flowing into the tank at a rate R(t), where R(t) is measured in gallons per minute, and t is measured in minutes. Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the number of gallons of water that are in the tank at time t = 11 minutes?

• (A) 40.6

• (B) 42.9

• (C) 75.6

• (D) 77.9