?1. Graph f(x) = 2x - 3.2. Starting with the graph in part (a), sketch a rough graph of

Find an antiderivative of the function. (a) \(f(x)=6\) (b) \(g(t)=3 t^{2}\) Equation Transcription: Text Transcription: f(x)=6 g(t)=3t^2

Find an antiderivative of the function. (a) \(f(x)=2 x\) (b) \(g(x)=-1 / x^{2}\) Equation Transcription: Text Transcription: f(x)=2x g(x)=-1/x^2

Find an antiderivative of the function. (a) \(h(q)=\cos q\) (b) \(f(x)=e^{x}\) Equation Transcription: Text Transcription: h(q)=cos q f(x)=e^x

Find an antiderivative of the function. (a) \(g(t)=1 / t\) (b) \(r(\theta)=\sec ^{2} \theta\) Equation Transcription: Text Transcription: g(t)=1/t r()=sec^2 theta?

Find the most general antiderivative of the function. (Check your answer by differentiation.) \(f(x)=4 x+7\) Equation Transcription: Text Transcription: f(x) = 4x + 7

Find the most general antiderivative of the function. (Check your answer by differentiation.) \(f(x)=x^{2}-3 x+2\) Equation Transcription: Text Transcription: f(x) = x^2 - 3x + 2

Find the most general antiderivative of the function. (Check your answer by differentiation.) \(f(x)=2 x^{3}-\frac{2}{3} x^{2}+5 x\) Equation Transcription: Text Transcription: f(x) = 2x^3 - 2/3x^2 + 5x

Find the most general antiderivative of the function. (Check your answer by differentiation.) \(f(x)=6 x^{5}-8 x^{4}-9 x^{2}\) Equation Transcription: Text Transcription: f(x) = 6x^5 - 8x^4 - 9x^2

Find the most general antiderivative of the function. (Check your answer by differentiation.) \(f(x)=x(12 x+8)\) Equation Transcription: Text Transcription: f(x) = x(12x + 8)

Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = (x - 5)2

Find the most general antiderivative of the function. (Check your answer by differentiation.) g(x) = 4x-? - 2x5/3

Find the most general antiderivative of the function. (Check your answer by differentiation.) h(z) = 3z0.8 + z-2.5

Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) =

Find the most general antiderivative of the function. (Check your answer by differentiation.) g(x) =

Find the most general antiderivative of the function. (Check your answer by differentiation.)

Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) =

Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) =

Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) =

Find the most general antiderivative of the function. (Check your answer by differentiation.) g(t) = 7et - e3

Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) =

Find the most general antiderivative of the function. (Check your answer by differentiation.) f(????) = 2 sin ???? - 3 sec ???? tan ????

Find the most general antiderivative of the function. (Check your answer by differentiation.) h(x) = sec2x + ???? cos x

Find the most general antiderivative of the function. (Check your answer by differentiation.) \(f(r)=\frac{4}{1+r^{2}}-\sqrt[5]{r^{4}}\)

Find the most general antiderivative of the function. (Check your answer by differentiation.) g(v) = 2 cos v -

Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 2x + 4 sinh x

Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) =

Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. \(f(x)=2 e^{x}-6 x\), F(0) = 1

Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) = 4 - 3(1 + x2)-1, F(1) = 0

Find f . f’’(x) = 24x

Find f . f’’(t) = t2 - 4

Find f . f’’(x) = 4x3 + 24x - 1

Find f . f’’(x) = 6x - x4 + 3x5

Find f . f’’(x) = 2x + 3ex

Find f . f’’(x) = 1/x2, x > 0

Find f . f’’’(t) = 12 + sin t

Find f . f’’’(t) =

Find f . f’(x) = 8x3 + , x > 0, f(1) = -3

Find f . f’(x) = - 2, f(9) = 4

Find f . f’(t) = 4/(1 + t2), f(1) = 0

Find f . f’(t) = t + 1/t3, t > 0, f(1) = 6

Find f . f’(x) = 5x2/3, f(8) = 21

Find f . \(\mathrm{f}^{\prime}(\mathrm{x})=(\mathrm{x}+1) / \sqrt{x}\), f(1) = 5

Find f . f’(t) = sec t (sec t + tan t), -????/2 < t < ????/2, f(????/4) = -1

Find f . f’(t) = 3t - 3/t, f(1) = 2, f(-1) = 1

Find f . f’’(x) = -2 + 12x - 12x2, f(0) = 4, f’(0) = 12

Find f . f’’(x) = 8x3 + 5, f(1) = 0, f’(1) = 8

Find f . f’’(????) = sin ???? + cos ????, f(0) = 3, f’(0) = 4

Find f. \(f^{\prime \prime}(t)=t^{2}+1 / t^{2}\), t > 0, f(2) = 3, \(f^{\prime}(1)=2\)

Find f . f’’(x) = 4 + 6x + 24x2, f(0) = 3, f(1) = 10

Find f . f’’(x) = x3 + sinh x, f(0) = 1, f(2) = 2.6

Find f . f’’(x) = ex - 2 sin x, f(0) = 3, f(????/2) = 0

Find f . f’’(t) = - cos t, f(0) = 2, f(1) = 2

Find f . f’’(x) = x-2, x > 0, f(1) = 0, f(2) = 0

Find f . f’’’(x) = cos x, f(0) = 1, f’(0) = 2, f’’(0) = 3

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 a function f such that f’(x) = x3 and the line x + y = 0 is tangent to the graph of f.

The graph of a function f is shown. Which graph is an antiderivative of f and why?

The graph of a function f is shown. Which graph is an antiderivative of f and why?

The graph of a function is shown in the figure. Make a rough sketch of an antiderivative F, given that F(0) = 1.

The graph of the velocity function of a particle is shown in the figure. Sketch the graph of a position function.

The graph of f’ is shown in the figure. Sketch the graph of f if f is continuous on [0, 3] and f(0) = -1.

1. Graph f(x) = 2x - 3. 2. Starting with the graph in part (a), sketch a rough graph of the antiderivative F that satisfies F(0) = 1. 3. Use the rules of this section to find an expression for F(x). 4. Graph F using the expression in part (c). Compare with your sketch in part (b).

Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin. f(x) = , -2???? ? x ? 2????

Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin. f(x) = , -3 ? x ? 3

A particle is moving with the given data. Find the position of the particle. v(t) = 2 cos t + 4 sin t, s(0) = 3

A particle is moving with the given data. Find the position of the particle. v(t) = t2 - 3, s(4) = 8

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. 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) = sin t - cos t, s(0) = 0, s(????) = 6

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 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 does it strike the ground? (d) If the stone is thrown downward with a speed of 5 m/s, how long does it take to reach the ground?

Show that for motion in a straight line with constant acceleration a, initial velocity \(\mathrm{v}_{0}\), and initial displacement \(\mathrm{s}_{0}\), the displacement after time t is \(\mathrm{s}=\frac{1}{2} a t^{2}+v_{0} t+s_{0}\).

An object is projected upward with initial velocity v0 meters per second from a point s0 meters above the ground. Show that

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?

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 ?

If a diver of mass m stands at the end of a diving board with length L and linear density , then the board takes on the shape of a curve y = f(x), where Ely’’ = mg(L - x) + (L - x)2 E and I are positive constants that depend on the material of the board and g(< 0) is the acceleration due to gravity. (a) Find an expression for the shape of the curve. (b) Use f(L) to estimate the distance below the horizontal at the end of the board.

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.

The linear density of a rod of length 1 m is given by p(x) = 1, in grams per centimeter, where x is measured in centimeters from one end of the rod. Find the mass of the rod.

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 mys and its downward acceleration is If the raindrop forms 500 m above the ground, how long does it take to fall?

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?

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 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?

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 multi-car pileup?

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?

A particular bullet train accelerates and decelerates at the rate of 2.4 ft/s2. Its maximum cruising speed is 180 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 that speed for 20 minutes? (b) Suppose that the train starts from rest and must come to a complete stop in 20 minutes. What is the maximum distance it can travel under these conditions? (c) Find the minimum time that the train takes to travel between two consecutive stations that are 60 miles apart. (d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?