Find f. f ’’(x)= -2 + 12x - 12x2, f (0) = 4, f ‘(0) = 12
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and t are increasing on an interval I, then f + g is increasing on I.
> Find y’ and y’’. y = ln (1 + ln x)
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. There exists a function f such that f (x) < 0, f ‘(x) < 0, and f ’’(x) > 0 for all x.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. There exists a function f such that f (x) > 0, f ‘(x) < 0, and f ’’ (x) > 0 for all x.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. There exists a function f such that f (1) = -2, f (3) = 0, and f ‘(x) > 1 for all x.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f ‘(x) = g’(x) for 0 < x < 1, then f (x) = g(x) for 0 < x < 1.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f ‘’(2) = 0, then (2, f (2)) is an inflection point of the curve y = f (x).
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f ‘(x) < 0 for 1 < x < 6, then f is decreasing on (1, 6).
> In this project we investigate the most economical shape for a can. We first interpret this to mean that the volume V of a cylindrical can is given and we need to find the height h and radius r that minimize the cost of the metal to make the can (see the
> Rainbows are created when raindrops scatter sunlight. They have fascinated mankind since ancient times and have inspired attempts at scientific explanation since the time of Aristotle. In this project we use the ideas of Descartes and Newton to explain t
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = x(12x + 8)
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 6x5 - 8x4 - 9x2
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 2x3 – 2/3x2 + 5x
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = x2 - 3x + 2
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 4x + 7
> A high-speed bullet train accelerates and decelerates at the rate of 4 ft/s2. Its maximum cruising speed is 90 mi/h. (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at th
> A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a(t) = 60t, at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (do
> A car is traveling at 100 km/h when the driver sees an accident 80 m ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup?
> A car braked with a constant deceleration of 16 ft/s2, producing skid marks measuring 200 ft before coming to a stop. How fast was the car traveling when the brakes were first applied?
> What constant acceleration is required to increase the speed of a car from 30 mi/h to 50 mi/h in 5 seconds?
> A car is traveling at 50 mi/h when the brakes are fully applied, producing a constant deceleration of 22 ft/s2. What is the distance traveled before the car comes to a stop?
> Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 m/s and its downward acceleration is If the raindrop is initially 500 m above the g
> A company estimates that the marginal cost (in dollars per item) of producing x items is 1.92 - 0.002x. If the cost of producing one item is $562, find the cost of producing 100 items.
> A stone was dropped off a cliff and hit the ground with a speed of 120 ft/s. What is the height of the cliff?
> Two balls are thrown upward from the edge of the cliff in Example 7. The first is thrown with a speed of 48 ft/s and the other is thrown a second later with a speed of 24 ft/s. Do the balls ever pass each other? Example 7: A ball is thrown upward with a
> Show that for motion in a straight line with constant acceleration a, initial velocity v0, and initial displacement s0, the displacement after time t is s = 1/2at2 + v0t + s0
> A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, 450 m above the ground. (a) Find the distance of the stone above ground level at time t. (b) How long does it take the stone to reach the ground? (c) With what velocity
> A particle is moving with the given data. Find the position of the particle. a(t) = t2 - 4t + 6, s(0) = 0, s(1) = 20
> A particle is moving with the given data. Find the position of the particle. a(t) = 10 sin t + 3 cos t, s(0) = 0, s(2π) = 12
> A particle is moving with the given data. Find the position of the particle. a(t) = 3 cos t - 2 sin t, s(0) = 0, v(0) = 4
> A particle is moving with the given data. Find the position of the particle. a(t) = 2t + 1, s(0) −= 3, v(0) = -2
> A particle is moving with the given data. Find the position of the particle. v(t) = sin t - cos t, s(0) = 0
> The graph of f ‘ is shown in the figure. Sketch the graph of f if f is continuous and f (0) = -1. y 2 y= f'(x) 1+ + 1 2 -1-
> The graph of the velocity function of a particle is shown in the figure. Sketch the graph of a position function. UA
> Differentiate the function. Y = log2 (x log5 x)
> Write the composite function in the form f ( g(x) ). [Identify the inner function u = g(x) and the outer function y = f (u).] Then find the derivative dy / dx. y = sin cot x
> The graph of a function is shown in the figure. Make a rough sketch of an antiderivative F, given that F(0) = 1. y. y = f(x)
> The graph of a function f is shown. Which graph is an antiderivative of f and why? f a X. b
> The graph of a function f is shown. Which graph is an antiderivative of f and why? y. f b a
> Find a function f such that f ,(x) = x3 and the line x + y = 0 is tangent to the graph of f .
> Given that the graph of f passes through the point (2, 5) and that the slope of its tangent line at (x, f (x)) is 3 - 4x, find f (1).
> Find f. f ’’’(x) = cos x, f (0) = 1, f ’’(0) = 2, f ’’’(0) = 3
> Find f. f ‘‘(x) = x-2, x > 0, f (1) = 0, f (2) = 0
> Find f. f ‘‘(x) = ex - 2sin x, f (0) = 3, f (π/2) = 0
> Find f. f ‘‘(x) = x3 + sinh x, f (0) = 1, f (2) = 2.6
> Find f. f ‘‘(x) = 4 + 6x + 24x2, f (0) = 3, f (1) = 10
> Find f. f ’’(t) = t2 + 1/t2, t > 0, f (2) = 3, f ‘(1) = 2
> Find f. f ’’(θ) = sin θ + cos θ, f (0) = 3, f ‘(0) = 4
> Find f. f ’’(x) = 8x3 + 5, f (1) = 0, f ‘(1) = 8
> Find f. f 9’(t) = 3t – 3/t , f (1) = 2, f (-1) = 1
> Find f. f ‘(x) = 5x2/3, f (8) = 21
> Find f. f ‘(t) = t + 1/t3, t > 0, f (1) = 6
> Find f. f ‘(t) = 4/(1 + t2), f (1) = 0
> Find f. f ‘(x) = 5x4 - 3x2 + 4, f (-) = 2
> Small birds like finches alternate between flapping their wings and keeping them folded while gliding (see Figure 1). In this project we analyze this phenomenon and try to determine how frequently a bird should flap its wings. Some of the principles are
> Differentiate the function. Y = ln (e-x + xe-x)
> Differentiate the function. Y = ln (csc x – cot x)
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f (x) = 1/5 – 2/x
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = e2
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 2
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 7x2/5 + 8x-4/5
> For which of the initial approximations x1 = a, b, c, and d do you think Newton’s method will work and lead to the root of the equation f (x) = 0? y. а b d
> For each initial approximation, determine graphically what happens if Newton’s method is used for the function whose graph is shown. (a) x1 = 0 (b) x1 = 1 (c) x1 = 3 (d) x1 = 4 (e) x1 = 5 y. + + 3 5
> Find the derivative of f (x) = (1 + 2x2)(x - x2) in two ways: by using the Product Rule and by performing the multiplication first. Do your answers agree?
> Suppose the tangent line to the curve y = f (x) at the point (2, 5) has the equation y = 9 - 2x. If Newton’s method is used to locate a root of the equation f (x) = 0 and the initial approximation is x1 = 2, find the second approximation x2.
> Follow the instructions for Exercise 1(a) but use x1 = 1 as the starting approximation for finding the root r. Data from Exercise 1: The figure shows the graph of a function f. Suppose that Newton’s method is used to approximate the ro
> The figure shows the graph of a function f. Suppose that Newton’s method is used to approximate the root s of the equation f (x) = 0 with initial approximation x1 = 6. (a) Draw the tangent lines that are used to find x2 and x3, and esti
> The figure shows the sun located at the origin and the earth at the point (1, 0). (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU ≈ 1.496 3 108 km.) There are five locations L1, L2, L3, L4, a
> In the figure, the length of the chord AB is 4 cm and the length of the arc AB is 5 cm. Find the central angle θ, in radians, correct to four decimal places. Then give the answer to the nearest degree. 5 cm A 4 cm
> Use Newton’s method to find the coordinates, correct to six decimal places, of the point on the parabola y = (x – 1)2 that is closest to the origin.
> Of the infinitely many lines that are tangent to the curve y = 2sin x and pass through the origin, there is one that has the largest slope. Use Newton’s method to find the slope of that line correct to six decimal places.
> Use Newton’s method to find the coordinates of the inflection point of the curve y = x2 sin x, 0 ≤ x ≤ π , correct to six decimal places.
> Use Newton’s method to find the absolute maximum value of the function f (x) = x cos x, 0 ≤ x ≤ π , correct to six decimal places.
> Differentiate the function. g(t) = 2t-3/4
> (a) Use Newton’s method to find the critical numbers of the function f (x) = x6 - x4 + 3x3 - 2x correct to six decimal places. (b) Find the absolute minimum value of f correct to four decimal places.
> If then the root of the equation f (x) = 0 is x = 0. Explain why Newton’s method fails to find the root no matter which initial approximation x1 ≠0 is used. Illustrate your explanation with a sketch. S(1) = {VE if
> (a) Use Newton’s method with x1 = 1 to find the root of the equation x3 - x = 1 correct to six decimal places. (b) Solve the equation in part (a) using x1 = 0.6 as the initial approximation. (c) Solve the equation in part (a) using x1 = 0.57. (You defini
> Explain why Newton’s method doesn’t work for finding the root of the equation x3 - 3x + 6 = 0 if the initial approximation is chosen to be x1 = 1.
> Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. cos (x2 - x) = x4
> Differentiate the function. T(z) = 2z log 2z
> Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. x5 - 3x4 + x3 - x2 - x + 6 = 0
> Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. -2x7 - 5x4 + 9x3 + 5 = 0
> Use Newton’s method to find all solutions of the equation correct to six decimal places. sin x = x2 - 2
> Use Newton’s method to find all solutions of the equation correct to six decimal places. 2x = 2 – x2
> Use Newton’s method to find all solutions of the equation correct to six decimal places. 3 cox x = x + 1
> Use Newton’s method to approximate the indicated root of the equation correct to six decimal places. The positive root of 3 sin x = x
> Use Newton’s method to approximate the indicated root of the equation correct to six decimal places. The negative root of ex = 4 - x2
> What is the maximum vertical distance between the line y = x + 2 and the parabola y = x2 for --1 ≤ x ≤ 2?
> (a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newton’s method to approximate the root correct to six decimal places. -2x5 + 9x4 - 7x3 - 11x = 0, [3, 4]
> The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares?
> (a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newton’s method to approximate the root correct to six decimal places. 3x4 - 8x3 + 2 = 0, [2, 3]
> Find two positive numbers whose product is 100 and whose sum is a minimum.
> Use Newton’s method to approximate the given number correct to eight decimal places. 8 500
> Find two numbers whose difference is 100 and whose product is a minimum.
> Use Newton’s method to approximate the given number correct to eight decimal places. 4 75
> Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the one at the right, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in
> Use Newton’s method with initial approximation x1 = 1 to find x2, the second approximation to the root of the equation x4 - x - 1 = 0. Explain how the method works by first graphing the function and its tangent line at (1, -1).
> Differentiate the function. F(s) = ln ln s
> Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line ,l, parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on , so that the i
> Use Newton’s method with initial approximation x1 = -1 to find x2, the second approximation to the root of the equation x3 + x + 3 = 0. Explain how the method works by first graphing the function and its tangent line at (-1, 1).
