Indefinite Integration Questions and Answers

Suppose the field has a length of 25,000 feet. You buy fencing to enclose it, but discover you've bought 10 feet too much. Rather than throwing away the extra fencing, you enclose a slightly larger field by enlarging each side of the field by the same amount, so the fencing encloses the field and a border: What is the largest animal that could be at point X and be entirely within the border region, touching neither the fence nor the field? (a) an ant, (b) a mouse, (c) a pigeon, (d) a cat, (e) an elephant. Find the average rate of change of the perimeter, if the length increases from x feet to x+h feet.

A maintenance worker fills a swimming pool with water at a rate of 10 m^3/minute. Suppose at t = 0, the swimming pool is filled with 100 m³ of water. How much water will be in the swimming pool at t = 17 minutes? ______m^3 A function giving the volume V of water in the swimming pool after t minutes is V(t) = ____________________. When will the swimming pool contain 430 m³ of water? ___________minutes.

Ken flies a drone in a circular path around an object that is 210 feet west and 190 feet north of his position.The drone's path takes it over a point that is 170 feet east and 250 feet south of him. Find an equation for the drone's path. (Assume Ken is located at the origin, with the horizontal axis running east-west and the vertical axis running north-south) The drone's path follows the equation_________ When the drone passes due west of Ken's position, it will be______ feet west of him (round your answer to three decimal places).

Evaluate the expression cos^-1(sin(4π/3)) ________________ Give your answer as an exact value

Suppose cos θ = 3/4 Find the following. cos(- θ) = cos (180° - θ) = cos(180° + θ) =

Let f(x) = x² - 2x - 2 for all x ≠ 3 The graph of f is given below Determine the following function value . Enter DNE if the function is undefined at x = 3 f(3) = _______

Find the length of an are of a circle of radius 18 feet intercepted by a central angle of 120°.

Read and understand the problem, then solve. Stating what is asked in the problem. During the Apollo 14 mission, Alan Shepard hit a golf ball on the Moon. The function h(t)=18t-0.8t² models the height of the golf ball's trajectory on the Moon, where h(t) is the height, in metres, of the ball above the surface of the Moon and t is the time in seconds. Determine the instantaneous rate of change in the height of the ball at 5 seconds.

The lines r = (2, 1, 1) + p(4, 0, -1), p∈ R, and q = (3,−1, 1) + q(9, −2, −2), q∈ R, intersect at the point A. Determine the coordinates of point A.

What is wrong with the equation? -2∫ 1 x^(-4)dx=(x^(-3)/(-3))| =-3/8

The graph of g is given below. Determine the intervals on which g is continuous. Enter the solution using interval notation. g is continuous on _____

U xx + U yy = 2x, 0 < x < 1, 0 < y < 1 u(x,0) = 0, u(1, y) = 0 u(x,0) = 0, u(x, 1) = 0 Solve your problem

Given that cosθ = 5/13 and that θ lies in Quadrant IV, what is the exact value of sin 2θ? A) 120 / 169 B) 24 / 13 C) -(24 / 13) D) -( 120 / 169)

Which of the following shows the simplified form of sinx/(1-cosx)? a)1 b) sin x + tan x c) sin x + cot x d) csc x + cot x

(This problem is similar to the example in your textbook about guessing a formula for the derivative of f(x) = x² and to a similar problem in the exercises. Study that example carefully before doing this problem.) Let f(t) = 2t² +4t. Estimate the following to within two decimal places by using small enough intervals. (a.) f'(1) ≈ (b.) f'(-5) = (c.) f'(9) ≈ (d.) Determine a formula for f'(t). f'(t) =

Solve the following system of equations. 3x+5y-5z-21 -4x=5z+17 2x-5y+6z=-16

Using the shift or stretch theorem find the Fourier transform of {1 for 2 ≤ t ≤0 b(t)= { 0, otherwise given the transform of unit square pulse function a(t)={1, for -1≤t≤1 { 0, otherwise is ā (k) =2sin(k)/k b(k)=___

Use the graph of f(x) given below to draw the graph of the function y = f(x + 3) - 2. Explain in words the changes in the graph.

Find two solutions of the equation. Give exact answers in radians in the interval [0,2π]. sin(t)=-(3^1/2)/2

Now let's write down an iterated double integral that is equal to ∬ e√(x² + y²) dA in polar coordinates in the dA = r dr dθ order.

Evaluate the sine, cosine, and tangent of the angle. Do not use a calculator. 405° sin(405°) = ______ cos(405°) = ______ tan(405°) = ______

Convert the degree measure to radian measure as a multiple of and as a decimal accurate to three decimal places. (-480°) multiple of π = ____ decimal = ____

Solve the following equation for over the interval [0, 2π), giving exact answers in radian units. If an equation has no solution, enter DNE. Multiple solutions should be entered as a comma-separated list. sin(4x) + 2 sin(2x) = 0 1)Use an appropriate double angle formula to rewrite the equaiton as: 2)Solve the equation for å over the interval [0, 2π):

Given functions m(x) = √x+12 and p(x) = x - 5, state the domains of the following functions using interval notation. Domain of m(x)/p(x): Domain of m(p(x)) : Domain of p(m(x)) :

Evaluate ∫ {(9x + 25) /x² +( 6x +9)} dx using the method of Partial Fractions.

Note: Triangle may not be drawn to scale. If ∠ B = 86° and a = 22, then Report lengths accurate to 4 decimal places. Question for thought: Can you obtain the final side without using the Pythagorean Theorem?

Ice Cream Sundaes. An ice cream shop has a choice of ten toppings. Suppose you can afford at most four toppings. How many different types of ice cream sundaes can you order?

Is 1800° coterminal with -1800°? (a) Yes (b) No

The marginal revenue (in thousands of dollars) from the sale of x gadgets is given by the following function. The revenue from 115 gadgets is $37,271. -2/3 MR(x) = 4x (= 4x (x² +28,000) (a) Find the revenue function. (b) What is the revenue from selling 250 gadgets? (c) How many gadgets must be sold for a revenue of at least $30,000? (a) R(x) = (Simplify your answer. Round to the nearest integer as needed.) (b) The revenue from selling 250 gadgets is $. (Type a whole number.) (c) To generate a revenue of at least $30,000, approximately (Type a whole number.)

Let f(x) = log3(x-4). a. Use transformations to graph the function. b. Write the domain and range in interval notation. c. Determine the vertical asymptote.

Evaluate the double integral for the function f(x, y) and the given region R. f(x, y) = y e^x^³; R is bounded by x =y/6, x = 1 and y = 0.

∫1-tan²x1+tan²x dx

Use Green's Theorem to evaluate the line integral along the given positiveley oriented curve: ∫x²y dx - 3y² dy, C is the circle x² + y² = 1

f and g are functions of x and y. f(1,2)= 2 and g(1,2)= 1. Also at (1,2) df = -2dx + 5dy dg = 4dx - 3dy z is a function of (x,y) defined implicitly by z2f+zg+x-y=0 with z(1, 2) = -1. Then dz dx(1, 2) = a) 1/2 b) -1/2 c) 5/3 d) -5/3 e) -2

Solve the equation for x. log (x+16) = log (x) + log (4) x=__

(a) Evaluate the following indefinite integrals. Show complete working. In particular: if you use substitution, write down the substitution that you used. Show the full working needed to make that substitution. Write your final answer in terms of a. if you use partial fraction decomposition, show the complete working that is needed to find the appropriate partial fraction expression. if you use integration by parts, indicate which part of the integration and corresponds to corresponds to v in this formula ∫du/dx vdx= uv =-∫u dv/dx dx (Note: the integrand is the function inside the integral that we want to integrate.) 1. ∫1/x²-1 dx 2. ∫ e^x cos h(3ex)dx 3. ∫ x ln xdx

Using the diagram on the right, choose the best estimate for the central angle in degrees. a) 263° c) 283° b) 273° d) 293°

Which x-value is in the domain of the function f (x) = 3cot(4x) + 2?

Two fair six-sided dice are rolled. Let A = the event that the dice add to 8, and let C = the event that the dice show the same number. Find P(CIA') and P(C'|A'). P(CIA')= (Simplify your answer.) P(C'IA')= (Simplify your answer.)

Solve by addition. 7x - 2y = 3 4x + 5y = 3.25

Find the slope of the graph of f(x) = ∜x when a) x = -1 b) x = 2 c) x = 0

Given the vector field F(x, y) = (x, xy, xyz) and the oriented line segment C from (1,1,1) to (2,1,0), compute c∫F. dr.

The terminal side of an angle 8 in standard position intersects the unit circle at(-12/13,-5/13) What is sin (θ)? Write your answer in simplified, rationalized form.

Solve the following equation by the quadratic formula below. 36x² + 7x-6=0 Give the answers in ascending order. Round your answers to three significant digits. x1 -0.51688 x2= 0.32244

Use a finite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 1) f(x) = x2 between x = 4 and x = 8 using the "midpoint rule" with four rectangles of equal width. 2) f(x) = x² between x = 0 and x = 1 using a lower sum with two rectangles of equal width.

Each side of a square is increasing at a rate of 6 cm/s. At what rate (in cm²/s) is the area cm²/s

f(x) = x² between x = 0 and x = 4 using an upper sum with two rectangles of equal width.

Find the total area of the region between the curve and the x-axis. y=1/√x; 1≤x≤ 4

Use the given transformation to evaluate the integral. ∫∫3x² dA, where R is the region bounded by the ellipse 9x^2 + 25y^2 = 225; x = 5u, y = 3v

Evaluate the integral by making an appropriate change of variables. ∫∫ 3 cos (9 (y-x / y+x)) dA where R is the trapezoidal region with vertices (8, 0), (10, 0), (0, 10), and (0, 8)