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mathematics
calculus

Questions and Answers of Calculus

Find a function f and a number a such that
The total cost of repaying a student loan at an interest rate of r % per year is C = f(r).(a) What is the meaning of the derivative f'(r)? What are its units?(b) What does the statement f'(10) = 1200
Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.
(a) If f(x) = √3 – 5x, use the definition of a derivative to find f’(x). (b) Find the domains of f and f’. (c) Graph f and f’ on a common screen compare the graphs to see whether your
(a) Find the asymptotes of the graph of and use them to sketch the graph.(b) Use your graph from part (a) to sketch the graph of f€™.(c) Use the definition of a derivative to find
The graph of f is shown. State, with reasons, the numbers at which f is not differentiable.
The total fertility rate at time t, denoted by F(t), is an estimate of the average number of children born to each woman(Assuming that current birth rates remain constant) The graph of the total
Let B(t) be the total value of U.S. banknotes in circulation at time . The table gives values of this function from 1980 to 1998, at year end, in billions of dollars. Interpret and estimate the value
Graph the curve y = (x + 1) / (x – 1) and the tangent lines to this curve at the points (2, 3) and (-1, 0)
Suppose that |f (x) | < g(x) for all x, where lim x→ a g(x) = 0. Find lim x →a f(x).
Let f(x) = [x] + [-x]. (a) For what values of does lim x→ a f(x) exist? (b) At what numbers is f discontinuous?
Evaluate lim x→1 3√x – 1 / √x – 1.
Find numbers a and b such that lim x→0 √ax + b – 2 / x = 1.
Evaluate lim x→0 |2x – 1| – |2x + 1| / x.
The figure shows a point P on the parabola y = x2 and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does
If [x] denotes the greatest integer function, find lim x→∞ x/[x].
Sketch the region in the plane defined by each of the following equations.(a) [x]2 + [y]2 = 1 (b) [x]2 – 1 [y]2 = 3 (c) [x + y]2 = 1 (d) [x] + [y] = 1
Find all values of a such that f is continuous on R:
A fixed point of a function f is a number in its domain such that f(c) = c. (The function doesn’t move ; it stays fixed.)(a) Sketch the graph of a continuous function with domain [0, 1] whose range
If lim x →a [f(x) + g(x)] = 2 and lim x →a [f(x) – g(x)] = 1, find lim x →a f(x)g(x).
(a) The figure shows an isosceles triangle ABC with
(a) If we start from 0o latitude and proceed in a westerly direction, we can let T(x) denote the temperature at the point at any given time. Assuming that T is a continuous function of , show that at
If f is a differentiable function and g(x) = xf(x), use the definition of a derivative to show that g’(x) = xf’(x) + f(x).
Suppose f is a function that satisfies the equation f(x + y) = f (x) + f(y) + x2y + xy2 for all real numbers x and y, suppose also that lim x →O f(x) / x = 1. (a) Find f(0). (b) Find
Suppose f is a function with the property that |f(x)| < x2 for all x. Show that f(0) = 0. Then show that f’(0) = 0.
(a) How is the number e defined?(b) Use a calculator to estimate the values of the limits correct to two decimal places.What can you conclude about the value of e?
(a) Sketch, by hand, the graph of the function f(x) = ex, paying particular attention to how the graph crosses the y-axis. What fact allows you to do this?(b) What types of functions are f(x) = ex
Find f ;( x) Compare the graphs of f and f€™ and use them to explain why your answer is reasonable.
Estimate the value of f€™ (a) by zooming in on the graph of f. Then differentiate to find the exact value of f€™ (a) and compare with your estimate.
Find an equation of the tangent line to the curve at the.
Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.
(a) Use a graphing calculator or computer to graph the function f(x) = x4 – 3x3 – 6x2 + 7x + 30 in the viewing rectangle [– 3, 5] by [– 10, 50].(b) Using the graph in part (a) to estimate
(a) Use a graphing calculator or computer to graph the function g(x) = ex – 3x2 in the viewing rectangle [– 1, 4] by.(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by
Find the points on the curve y = 2x3 + 3x2 – 12x + 1 where the tangent is horizontal.
For what values of does the graph of f(x) = x3 + 3x2 + x + 3 have a horizontal tangent?
Show that the curve y = 6x3 + 5x – 3 has no tangent line with slope 4.
At what point on the curve y = 1 + 2ex – 3x is the tangent line parallel to the line 3x – y = 5? Illustrate by graphing the curve and both lines.
Draw a diagram to show that there are two tangent lines to the parabola y = x2 that pass through the point (0, - 4). Find the coordinates of the points where these tangent lines intersect the
Find equations of both lines through the point (2, – 3) that are tangent to the parabola y = x2 + x.
A particle moves according to a law of motion s = f (t) t > 0, where is measured in seconds and in feet.(a) Find the velocity at time t.(b) What is the velocity after 3 s?(c) When is the particle at
The position function of a particle is given byWhen does the particle reach a velocity of 5 m/s?
If a ball is given a push so that it has an initial velocity of 5 m/s down a certain inclined plane, then the distance it has rolled after seconds is s = 5t + 3t2.(a) Find the velocity after 2 s.(b)
If a stone is thrown vertically upward from the surface of the moon with a velocity of 10 m/s, its height (in meters) after t seconds is h = 10t – 0.83t2.(a) What is the velocity of the stone after
If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after seconds is s = 80t – 16t2.(a) What is the maximum height reached by the ball?(b) What is the velocity of the
(a) A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area A(x) of a wafer changes when the
(a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x,
(a) Find the average rate of change of the area of a circle with respect to its radius as changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when r =
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after (a) 1 s, (b) 3 s, and(c) 5
A spherical balloon is being inflated. Find the rate of increase of the surface area (S = 4πr2 with respect to the radius when is? (a) 1 ft, (b) 2 ft, and (c) 3 ft. What conclusion can you
(a) The volume of a growing spherical cell is V = 4/3πr, where the radius is measured in micrometers (1 μ m = 10-6 m). Find the average rate of change of with respect to when changes from
The mass of the part of a metal rod that lies between its left end and a point meters to the right is 3x2 kg. Find the linear density (see Example 2) when is? (a) 1 m, (b) 2 m, and(c) 3 m. Where is
If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli€™s Law gives the volume V of water remaining in the tank after minutes asFind
The quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time t (measured in seconds) is given by Q(t) = t3 – 2t2 + 6t + 2. Find the current when(a) t = 0.5 s and
Newton€™s Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is where G is the gravitational constant and is the distance between the
Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV = C. (a) Find the rate of change of volume
The data in the table concern the lactonization of hydroxyvaleric acid at 25oC. They give the concentration C(t)of this acid in moles per liter after minutes.(a) Find the average rate of reaction for
The table gives the population of the world in the 20th century.(a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines.(b) Use a graphing
The table shows how the average age of first marriage of Japanese women varied in the last half of the 20th century.(a) Use a graphing calculator or computer to model these data with a fourth-degree
If, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value [A] =
Suppose that a bacteria population starts with 500 bacteria and triples every hour.(a) What is the population after 3 hours? After 4 hours after hours?(b) Use (5) in Section 3.1 to estimate the rate
Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynes/cm2 , and viscosity η = 0.027. (a) Find the
The frequency of vibrations of a vibrating violin string is given by where L is the length of the string,T is its tension, and is its linear density. [See Chapter 11 in Donald E. Hall,
Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is(a) Find the marginal cost function.(b) Find C'(100) and explain its meaning. What does it predict?(c)
The cost function for a certain commodity is C(x) = 84 + 0.16X - 0.0006X2 + 0.000003X3(a) Find and interpret C'(100).(b) Compare C'(100) with the cost of producing the 101st item.
If p(x) is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is(a) Find A'(x). Why does the company want to hire more
If R denotes the reaction of the body to some stimulus of strength, the sensitivity is defined to be the rate of change of the reaction with respect to x. A particular example is that when the
The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV = nRT, where is the number of moles of the gas and R = 0.0821 is the
In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation where r0 is the birth rate of
In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by W(t), and caribou, given by C(t), in northern
Prove, using the definition of derivative, that if f(x) = cos x, then f’(x) = – 1 sin x.
Find an equation of the tangent line to the curve at the given point.
(a) Find an equation of the tangent line to the curve y = x cos x at the point (π, – π). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) Find an equation of the tangent line to the curve y = x sec x – 2 cos x at the point (π/3, 1). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) If f(x) = 2x + cot x, find f’ (x). (b) Check to see that your answer to part (a) is reasonable by graphing both f and f’ for 0 < x < π.
(a) If f(x) = √x sin x, find f’ (x). (b) Check to see that your answer to part (a) is reasonable by graphing both f and f’ for 0 < x < 2π.
For what values of does the graph of f (x) = x + 2 sin x have a horizontal tangent?
Find the points on the curve y = (cos x) / (2 + sin x) at which the tangent is horizontal.
A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x(t) = 8 sin t, where is in seconds and in centimeters.(a) Find the velocity at time
An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s = 2 cos t +
A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let be the distance from the bottom of the ladder to the wall. If the bottom
An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is where
Differentiate each trigonometric identity to obtain a new (or familiar) identity.
A semicircle with diameter PQ sits on an isosceles triangle to PQR form a region shaped like an ice-cream cone, as shown in the figure. If A(θ) is the area of the semicircle and B(θ)is
The figure shows a circular arc of length and a chord of length d, both subtended by a central angle θ. Find
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.
Find an equation of the tangent line to the curve at the given point.
(a) Find an equation of the tangent line to the curve y = 2 / (1 + e–x) at the point (0, 1).(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) The curve y = |x|/√2 – x2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1). (b) Illustrate part (a) by graphing the curve and the
(a) If f(x) = √1 – x2/x, find f’(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f’.
The function f(x) = sin (x + sin 2x), 0 < x < π, arises in applications to frequency modulation (FM) synthesis. (a) Use a graph of f produced by a graphing device to make a rough sketch of the
Find all points on the graph of the function f(x) = 2 sin x + sin2x at which the tangent line is horizontal.
Find the x-coordinates of all points on the curve y = sin 2x – 2 sin x at which the tangent line is horizontal.
Suppose that F(x) = f(g(x)) and g(3) = 6, g’(3) = 4, f’(3) = 2, and f’(6) = 7, Find F’(3).
Suppose that w = u o v and u(0) = 1, v(0) = 2, u’(0) = 3, u’(2) =4, v’(0) = 5, and v’(2) = 6. Find w’(0).
A table of values for f, g, f€™, and g€™ is given.(a) If h(x) = f(g(x)), find h€™(1).(b) If H(x) = g(f(x)), find H€™(1).
Let f and g be the functions in Exercise 55.(a) If F(x) = f(f(x)), find F’(2).(b) If G(x) = g(g(x)), find G’(3)
If f and g are the functions whose graphs are shown, let u(x) = f (g(x)), v(x) = g(f(x)), and w(x) = g(g(x)). Find each derivative, if it exists. If it does not exist, explain why.(a) u€™
If f is the function whose graph is shown, let h(x) = f(f(x)) and g(x) = f(x2). Use the graph of f to estimate the value of each derivative.(a) h€™ (2)(b) g€™(2)
Use the table to estimate the value of h€™ (0, 5), where h(x) = f (g(x)).
If g(x) = f(f(x)), use the table to estimate the value of g€™(1)

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