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 (downward) velocity slows linearly to -18 ft/s in 5 seconds. The rocket then “floats” to the ground at that rate. (a) Determine the position function s and the velocity function v (for all times t). Sketch the graphs of s and v. (b) At what time does the rocket reach its maximum height, and what is that height? (c) At what time does the rocket land?
> Find the derivative of the function. f(x) = (2x - 3)4 (x2 + x + 1)5
> If h(2) = 4 and h’(2) = -3, find d h(x) dx I-2
> Differentiate f and find the domain of f. f(x) = ln ln ln x
> Find the derivative of the function. f(t) = eat sin bt
> Find the derivative of the function. f(t)= t sin πt
> Find the derivative of the function. Y = x2 e-3x
> Find the derivative of the function. g (θ) = cos2 (θ)
> Find the derivative of the function. f (θ) = cos (θ2)
> Suppose that f(4) = 2, g(4) = 5, f ‘ (4) = 6 and g’ (4) = -3. Find h ‘ (4). (a) h(x) = 3f (x) + 8g(x) (b) h(x) = f(x) g(x) (c) h(x) = f(x) / g(x) (d) h(x) = g(x) / f(x) + g(x)
> Find the derivative of the function. F(x) = (1 + x + x2)99
> Find the derivative of the function. F(x) = (5x6 + 2x3)4
> If f (x) = ex g(x), where g(o) = 2 and g’(o) = 5, find f ‘(0).
> Differentiate f and find the domain of f. f(x) = ln (x2 – 2x)
> Differentiate. y = 2 sec x - csc x
> Differentiate. f (x) = ex cos x
> If f is a differentiable function, find an expression for the derivative of each of the following functions. (a) y = x2 f (x) (b) y = f (x) / x2 (c) y = x2 / f (x) (d) y = 1 + x f (x) / √x
> Differentiate. f (x) = x cos x + 2 tan x
> Find the 50th derivative of y=cos 2x.
> Differentiate the function. H(u) = (3u - 1)(u + 2)
> Differentiate. f (x) = (3x2 - 5x)ex
> Suppose that f (5) = 1, f ‘(5) = 6, g(5) = -3, and g’ (5) = 2. Find the following values. (a) (fg) ‘ (5) (b) (f/g) ‘ (5) (c) (g/f) ‘ (5)
> 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 is differentiable and f (-1) = f (1), then there is a number c such that |c | < 1 and f ‘(c) =
> 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 is continuous on (a, b), then f attains an absolute maximum value f (c) and an absolute minimu
> 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 has an absolute minimum value at c, then f ‘(c) = 0.
> 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 ‘(c) = 0, then f has a local maximum or minimum at c.
> 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) exists and is nonzero for all x, then f (1) ≠ f (0).
> 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. The most general antiderivative of f sxd − x22 is F(x) = -1/x + C
> 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 is periodic, then f ’ is periodic.
> 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 is even, then f ’ is even.
> 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 is increasing and f (x) > 0 on I, then g(x) = 1/f (x) is decreasing on I.
> 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 positive increasing functions on an interval I, then f g is increasing on I.
> 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.
> 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.
> 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 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 ’’(x)= -2 + 12x - 12x2, f (0) = 4, f ‘(0) = 12
> 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
