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mathematics
calculus

Questions and Answers of Calculus

For the given cost and demand functions, find the production level that will maximize profit.
Find the production level at which the marginal cost function starts to increase.
The cost, in dollars, of producing yards of a certain fabric is C(x) = 1200 + 12x – 0.1x2 + 0.0005x3 and the company p(x) = 29 – 0.00021x finds that if it sells yards, it can charge dollars per
An aircraft manufacturer wants to determine the best selling price for a new airplane. The company estimates that the initial cost of designing the airplane and setting up the factories in which to
A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the average attendance had been 27,000. When ticket prices were lowered to $8, the average attendance rose
During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found
A manufacturer has been selling 1000 television sets a week at $450 each. A market survey indicates that for each $10 rebate offered to the buyer, the number of sets sold will increase by 100 per
The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $800 per month. A market survey suggests that, on average, one additional unit will
Store managers want an optimal inventory policy. Overstocking results in excessive storage and interest costs, whereas a small inventory means added costs for reordering and delivery. A supermarket
Suppose a person has an amount A of spending money deposited to her savings account each month, where it earns interest at a monthly rate R. Assume she spends the entire amount throughout the month
The figure shows the graph of a function f, suppose that Newton€™s method is used to approximate the root of the equation f(x) = 0 with initial approximation x1 = 1.(a) Draw the tangent
Follow the instructions for Exercise 1(a) but use s1 = 9 as the starting approximation for finding the root.
Suppose the line y = 5x – 4 is tangent to the curve y = f(x) when x = 3. If Newton’s method is used to locate a root of the equation f(x) = 0 and the initial approximation is x1 = 3, find the
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(a) x1 = 5
Use Newton€™s method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places.)
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
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
Use Newton€™s method to approximate the given number correct to eight decimal places.
Use Newton€™s method to approximate the indicated root of the equation correct to six decimal places.
Use Newton’s method to find all roots of the equation correct to six decimal places.
Use Newton€™s method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations.
(a) Apply Newton€™s method to the equation x2 €“ a = 0 to derive the following square-root algorithm used by the ancient Babylonians to compute √ a):(b) Use part (a) to
(a) Apply Newton’s method to the equation 1/x – a = 0 to derive the following reciprocal algorithm: xn+1 = 2xn – ax2n (This algorithm enables a computer to find reciprocals without actually
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.
(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)
Explain why Newton’s method fails when applied to the equation 3√x = 0 with any initial approximation x ≠ 0. Illustrate your explanation with a sketch.
IfThen 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
(a) Use Newton’s method to find the critical numbers of the function f(x) = 3x4 – 28x3 + 6x3 + 24x correct to three decimal places. (b) Find the absolute minimum value of the function f(x) = 3x4
Use Newton’s method to find the absolute minimum value of the function f(x) = x2 + sin x correct to six decimal places.
Use Newton’s method to find the coordinates of the inflection point of the curve y = e cos x, 0, < x < π correct to six decimal places.
Of the infinitely many lines that are tangent to the curve y = – sin 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
A grain silo consists of a cylindrical main section, with height 30 ft, and a hemispherical roof. In order to achieve a total volume of 15,000 ft3 (including the part inside the roof section), what
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
A car dealer sells a new car for $18,000. He also offers to sell the same car for payments of $375 per month for five years. What monthly interest rate is this dealer charging? To solve this problem
The figure shows the Sun located at the origin and Earth at the point (1, 0).(The unit here is the distance between the centers of Earth and the Sun, called an astronomical unit: 1 AU ≈ 1.496 X
Given that the graph of f passes through the point (1, 6) and that the slope of its tangent line at (x, f(x)) is 2x + 1, find f (2).
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 anti-derivative of f and why?
The graph of a function is shown in the figure. Make a rough sketch of an anti-derivative, given that F (0) = 0.
The graph of the velocity function of a car is shown in the figure. Sketch the graph of the position function.
The graph of f€™ is shown in the figure. Sketch the graph of f if f is continuous and
(a) Use a graphing device to graph f(x) = 2x – 3√x. (b) Starting with the graph in part (a), sketch a rough graph of the anti-derivative F that satisfies F(0) = 1. (c) Use the rules of this
Draw a graph of f and use it to make a rough sketch of the anti-derivative that passes through the origin.
A direction field is given for a function. Use it to draw the anti-derivative F that satisfies F(0) = - 2
Use a direction field to graph the anti-derivative that satisfies F(0) = 0.
A function is defined by the following experimental data. Use a direction field to sketch the graph of its anti-derivative if the initial condition is F(0) =0.
(a) Draw a direction field for the function f(x) = 1/x2 and use it to sketch several members of the family of anti-derivatives.(b) Compute the general anti-derivative explicitly and sketch several
A particle is moving with the given data. Find the position of the particle.
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
Show that for motion in a straight line with constant acceleration a, initial velocity vo, and initial displacement so the displacement after time t is s = ½ at2 + v0t + s0.
An object is projected upward with initial velocity vo meters per second from a point so meters above the ground. Show that
Two balls are thrown upward from the edge of the cliff in Example 8. The first is thrown with a speed of 48ft/s and the other is thrown a second later with a speed of 24ft/s. Do the balls ever pass
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 and E and are positive constants that depend
A company estimates that the marginal cost (in dollars per item) of producing 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 m is given by p(x) = 1/√x, in grams per centimeter, where 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 10m/s and its downward
A car is traveling at 50 mi/h when the brakes are fully applied, producing a constant deceleration of 22 ft/s. What is the distance covered 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 s?
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 100km/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 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.
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
Explain the difference between an absolute maximum and a local maximum. Illustrate with a sketch.
If f’ (c) = 0, then f has a local maximum or minimum at c.
If f has an absolute minimum value at c, then f(c) = 0.
If f is continuous on (a, b), then f attains an absolute maximum value f (c) and an absolute minimum value f (d) at some numbers c an d in (a, b).
If f is differentiable and f (-1) = f (1), then there is a number c such that | c | < 1 and f’ (c) = 0.
If f’(x) < 0 for a < x < 6, then f is decreasing on (1, 6).
If f” (2) = 0, then (2, f (2)) is an inflection point of the curve y = f(x).
If f’(x) = g’(x) for 0 < x < 1, then f’(x) = g(x) for 0 < x
There exists a function f such that f (1) = – 2, f(3) = 0, and f’(x) > 1 for all x.
There exists a function f such that f(x) > 0, f’(x) < 0, and f” (x) > 0 for all x.
There exists a function f such that f(x) < 0, f’(x) < 0, and f”(x) > 0 for all x.
If f and g are increasing on an interval l, then f + g is increasing on l.
If f and g are increasing on an interval l, then f – g is increasing on l.
If f and g are increasing on an i9nterval l, then fg is increasing on l.
If f and g are positive increasing functions on an interval l, then fg is increasing on l,
If f is increasing and f(x) > 0 on l, the g(x) = 1/f(x) is decreasing on l.
The most general anti-derivative of f(x) = x-2 is F(x) = – 1 /x + c
If f’(x) exists and is nonzero for all x, then f (1) ≠ f (0).
Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f€™ and f€ to estimate the intervals of increase and decrease, extreme values, intervals of
Graph f(x) = e-1/x2 in a viewing rectangle that shows all the main aspects of this function. Estimate the inflection points. Then use calculus to find them exactly.
(a) Graph the function f(x) = 1 / (1 + e1/x). (b) Explain the shape of the graph by computing the limits of f(x) as approaches ∞, – ∞, 0+, and 0–. (c) Use the graph of f to estimate
If f(x) = arc tan (cos (3 arc sin x)), use the graphs of f, f’, and f” to estimate the x-coordinates of the maximum an minimum points and inflections points of f.
If f(x) = in (2x + x sin x), use the graphs of f, f”, and f” to estimate the intervals of increase and the inflection point of f on the interval (0, 15).
Investigate the family of functions f(x) = in (sin x + C). What features fo the members of this family have in common? How do they differ? For which values of C is f continuous on (– ∞,
Investigate the family of functions f(x) = cxe–cx2. What happens to the maximum and minimum points and the inflection points as c changes? Illustrate your conclusions by graphing several members of
Show that the equation x101 + x 51 + x – 1 = 0 has exactly one real root.
Suppose that f is continuous on [0, 4], f (0) = 1, and 2 < f’(x) < 5 for all x in (0, 4). Show that 9 < f (4) < 21.
By applying the Mean Value Theorem to the function f (x) = x1/5 on the interval [32, 33], show that 2
For what values of the constants a and b is (1, 6) a point of inflection of the curve y = x3 + ax2 + bx + 1?
Let g(x) = f(x) 2, where f is twice differentiable for all x, f’(x) > 0 for all x ≠ 0, and f is concave downward on (– ∞, 0) and concave upward on (0, ∞). (a) At what numbers
Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible.
Show that the shortest distance from the point (x1, y1) to the straight line Ax + By + C = 0 is
Find the point on the hyperbola xy = 8 that is closest to the point (3, 0)
Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r.
Find the volume of the largest circular cone that can be inscribed in a sphere of radius r.
In ∆ABC, D lies on AB, CD ┴ AB, | AD | = | B D | = 4 cm, AND | CD | = 5 cm. Where a pint P should be chosen on CD so that the sum |PA| + |PB| + |PC| are a minimum?
Do Exercise 55 when | CD | = 2 cm.

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