Definite Integrals Questions and Answers

Find f'(x) and find the value(s) of x where the tangent line is horizontal. f(x)=x6(x-13)7 f'(x) = Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The tangent line is horizontal at x = (Use a comma to separate answers as needed.) B. The tangent line is never horizontal.

Find the area of the region bounded by the x-axis and the graph of the function. f(x)= = x/(6 + x²)^2

Find the area of the region under the curve y = f(x) over the indicated interval. f(x) = 4 (x + 5)^3 X 20

Find the amount of a continuous money flow in which 1500 per year is being invested at 5.5%, compounded continuously for 40 years. Round the answer to the nearest cent. A. $218,864.00 B. $246,136.73 C. $1,203,752.02 D. $273,409.46

Partial Fraction Decomposition (x² - 11x-18)/x(x²+3x+3)

Find the accumulated future value of the continuous income stream at rate R(t), for the given time T, and interest rate k, compounded continuously. R(t) = $500,000, T = 17 years, k=6%

Find the accumulated present value of the following continuous income stream at rate R(t), for the given time T and interest rate k, compounded continuously. R(t) = 0.04t+100, T=10, k= 4%

Solve for t to two decimal places. 6 = e^ 0.05t

Evaluate, in spherical coordinates, the triple integral of f(ρ, θ, Φ) = cos , over the region 0 ≤ θ ≤ 2π, π/3 ≤ Φ ≤ π/2,3 ≤ ρ,≤ 6.

Factor the polynomial completely. [Hint: Do not use space in your answer.] m³ - 125

Evaluate the integral sinx from 0 to pi/2 by 1) using trapezoidal rule 2) Simpson rule Using 11 ordinates. Also estimate the errors by finding the value of the integral

A company estimates that its sales will grow continuously at a rate given by the function S'(t) = 15 e', where S'(t) is the rate at which sales are increasing, in dollars per day, on day t. Find the sales from the 2nd day through the 8th day. (This is the integral from 1 to 8.) A. $44,714.37 B. $44,673.60 C. $20,866.71 D. $2,978.24

Find the area bounded by the given curves. Round to two decimal places. x=0, x= -2₁ y=e^x, y=0 A. 0.96 B. 0.86 C. 0.68 D. 0.78

Find the area of the shaded region. f(x)=x^4-8x³ +21x², g(x) = 6x+36

Approximate the area under the graph of f(x) = 0.05x^4 -3.61x² +95 over the interval [5,9] by dividing the interval into 4 subintervals.

Find the area under the graph of f over the interval [1,9]. f(x)= 8x +9, for x ≤3 48 - 5/2x, for x>3

A company determined that the marginal cost, C'(x) of producing the xth unit of a product is given by C'(x)=x4-2x. Find the total cost function C, assuming that C(x) is in dollars and that fixed costs are $4000.

Given y = 5x² + 2x, find dy/dt when x = -1 and dx/dt = 3.

Sin t and cost are given. Use identities to find the indicated value. Where necessary, rationalize denominators. sin t =2/7, cos t = -3√5/7. Find tan t.

Let C be the curve of intersection of the parabolic cylinder x² = 2y, and the surface 3z = xy. Find the exact length of C from the origin to the point (5, 25/2 125/6)

Find the two x-intercepts of f(x)=√x and show that f'(x) = 0 at some point between the two x-intercepts. Determine whether Rolle's Theorem can be applied to f on the closed interval [0, 8]. If Rolle's Theorem can be applied, find all values of c in the open interval (0, 8) such that f'(c) = 0.

d the area (in square units) of the region under the graph of the function f on the interval [-1, 2]. f(x) = -x² + 5 2 square units

Find the area (in square units) of the region under the graph of the function f on the interval [1, 2], using the Fundamental Theorem of Calculus. Then verify your result using geometry. f(x) - 1 -x+1 2 square units

Find the area of the region under the graph of the function f on the interval [1¹, 5]. f(x) = 4x³ square units

Find the area between the curves y = e^ -0.1x and y = 1.2x + 1 from x= 0 to x = 5.

Find c > 0 such that the area of the region enclosed by the parabolas y = x² - c²and y=-c²-x² is 300.

Which statement is true? There are many derivatives of a function g(x). There is only one anti-derivative, f(x), for a function g(x). There are many anti-derivatives, f(x), of a function g(x). There are only two anti-derivative f(x), for a function g(x).

An object with an initial velocity of v(0) = 3 has an acceleration of a(t) = 5 + 6t. What is its velocity at t = 5? 35 97 100 103

Evaluate -2∫ 1 (x-2)(x + 3) dx 53/3 33/2 -32/3 -33/2

What is the average value of f(x) between x = 0 and x = 4 for f(x) = 2x - 1? A. 6 B. 3 C. 0 D. -3

An object is moving along a horizontal line such that it's velocity is v(t) = 3/(3t+4) What is the object's displacement between t = 0 and t = 5? A. 1.558 B. 1.944 C. 2.558 D. 2.944

Find the total change of f(x) between x = 0 and x = 3, when df/dx = 3/x+4 A. 0.56 B. 1.679 C. 3.332 D. 9.997

What is the total change of f(x), if f'(x) = 3x + 1, over the interval [1, 5]? A. -40 B. 40 C. 42.5 D. 45

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's rule to approximate the integral ₁∫³ ln(x)/3+x dx with n = 8.

If 0<k<π/2 and the area under the curve y=cosx from x=k to x=π/2 is 0.1, then k= (A) 1.471 (B) 1.414 (C) 1.277 (D) 1.120 (E) 0.436

The distance x, in feet, between successive cars on a certain stretch of highway has the following probability density function, where k = 1/a and a is the average distance, in feet, between successive cars over some period of time. f(x)=ke-kx, for 0 ≤ x <∞ A transportation planner determines that the average distance between cars on a certain highway is 130 ft. What is the probability that the distance between two successive cars, chosen at random, is 45 ft or less? P(0≤x≤45)= (Round the final answer to four decimal places as needed. Round all intermediate values to four decimal places as needed.)

*4. Use the comparison theorem for improper integrals to prove that S verges. sin r 1³ Hint: Use your answer from question 3(a) combined with the fact that sin r ≤ 1. de con-

2. Using the disk method, find the volume of the solid of revolution formed by revolving and the z-axis over the interval [1, 2] around the x-axis. 1 the region bounded y = √x(3-x) Use graphing software or a graphing calculator to help see the situation.