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mathematics
calculus

Questions and Answers of Calculus

A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building
A man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of . At what rate are the people moving apart 15 min after
A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s.(a) At what rate is his distance from second base decreasing when he is
The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min, at what rate is the base of the triangle changing when the altitude
A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1
At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 P.M.?
A particle is moving along the curve y = x. As the particle passes through the point (4, 2), its -coordinate increases at a rate of 3 cm/s. How fast is the distance from the particle to the origin
Water is leaking out of an inverted conical tank at a rate of 10,000 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at
A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft3/min
A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled
A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A cross-section is shown in the figure. If the pool is being filled at a rate of
Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is
A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string have been let out?
Two sides of a triangle are 4 m and 5 m in length and the angle between them is increasing at a rate of 0.06 rad/s. Find the rate at which the area of the triangle is increasing when the angle
Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2o/min. How fast is the length of the third side increasing when the angle between the sides of
Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain
When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV1.4 = C, where C is a constant. Suppose that at a certain instant the volume
If two resistors with resistances R1 and R2 are connected in parallel, as in the figure, then the total resistance R, measured in ohms (Ω), is given by 1/R = 1/R1 + 1/R2 If R1 and R2 are
Brain weight B as a function of body weight W in fish has been modeled by the power function B = 0.007W2/3, where B and W are measured in grams. A model for body weight as a function of body length L
A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2 ft/s, how fast is the angle between the top of the ladder and the wall
Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the floor 12 ft directly beneath P and between the carts. Cart A is being pulled
A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also,
A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the
A plane flying with a constant speed of 300 km/h passes over a ground radar station at an altitude of 1 km and climbs at an angle of 30o. At what rate is the distance from the plane to the radar
Two people start from the same point. One walks east at 3 mi/h and the other walks northeast at 2 mi/h. How fast is the distance between the people changing after 15 minutes?
A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runner’s friend is standing at a distance 200 m from the center of the track. How fast is the distance
The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o’clock?
The turkey in Example 1 is removed from the oven when its temperature reaches 185oF and is placed on a table in a room where the temperature is 172oF. After 10 minutes the temperature of the turkey
Atmospheric pressure P decreases as altitude h increases. At a temperature of 15oC, the pressure is 101.3 kilopascals (kPa) at sea level, 87.1 kPa at h = 1 km, and 74.9 kPa at h = 2 km. Use a linear
The graph indicates how Australia€™s population is aging by showing the past and projected percentage of the population aged 65 and over. Use a linear approximation to predict the
The table shows the population of Nepal (in millions) as of June 30 of the given year. Use a linear approximation to estimate the population at midyear in 1984. Use another linear approximation to
Find the linearization L(x) of the function at a.
Find the linear approximation of the function f(x) = √1 – x at a = 0 and use it to approximate the numbers √0.9 and √0.9. Illustrate by graphing f and the tangent line.
Find the linear approximation of the function g(x) = 3√1 + x at a = 0 and use it to approximate the numbers 3√0.95 and 3√1.1. Illustrate by graphing and the tangent line.
Verify the given linear approximation at a = 0. Then determine the values of for which the linear approximation is accurate to within 0.1.
(a) Find the differential dy and(b) Evaluate dy for the given values of and dx.
Compute ∆y and dy for the given values of and dx = ∆x. Then sketch a diagram like Figure 6 showing the line segments with lengths dx, dy, and ∆y.
Use differentials (or, equivalently, a linear approximation) to estimate the given number.
Explain, in terms of linear approximations or differentials, why the approximation is reasonable.
Let f(x) = (x – 1)2 g(x) = e –2x and h(x) = 1 + in (1 – 2x)(a) Find the linearization’s of f, g, and h at a = 0. What do you notice? How do you explain what happened? (b) Graph f, g, and
The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing
The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm.(a) Use differentials to estimate the maximum error in the calculated area of the disk.(b) What is the
The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm.(a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative
Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m.
Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule.
When blood flows along a blood vessel, the flux F (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel: F = kR4.
Establish the following rules for working with differentials (where denotes a constant and u and v are functions of x).
On page 431 of Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/Cole, 2000), in the course of deriving the formula T = 2π√L/g for the period of a pendulum of length L,
Suppose that the only information we have about a function is that f (1) = 5 and the graph of its derivative is as shown.(a) Use a linear approximation to estimate f (0.9) and f (1.1).(b) Are your
Suppose that we don’t have a formula g(x) for but we know that g(2) = – 4 and g’(x) = √x2 + 5 for all x. (a) Use a linear approximation to estimate g(1.95) and g(2.05). (b) Are your
Find the quadratic approximation P(x) = A + Bx + Cx2 to the function f(x) = cos x that satisfies conditions (i), (ii), and (iii) with a = 0. Graph P, f, and the linear approximation L(x) = 1 on a
Determine the values of for which the quadratic approximation f(x) = P(x) in Problem 1 is accurate to within 0.1.
To approximate a function f by a quadratic function P near a number a, it is best to write P in the form P(x) = A + B(x – a) + C(x – a) 2, show that the quadratic function that satisfies
Find the quadratic approximation to f(x) = √x + 3 near a = 1. Graph f, the quadratic approximation, and the linear approximation from Example 3 in Section 3.11 on a common screen. What do you
Find a cubic polynomial P(x) = ax3 + bx2 + cx + d that satisfies condition (i) by imposing suitable conditions on P(x) and P’(x) at the start of descent and at touchdown.
Use conditions (ii) and (iii) to show that
Suppose that an airline decides not to allow vertical acceleration of a plane to exceed k = 860 mi/h2. If the cruising altitude of a plane is 35,000 ft and the speed is 300 mi/h, how far away from
Graph the approach path if the conditions stated in Problem 3 are satisfied.
(a) Suppose the horizontal distance between P and Q is 100 ft. Write equations in a, b, and c that will ensure that the track is smooth at the transition points(b) Solve the equations in part (a) for
The solution in Problem 1 might look smooth, but it might not feel smooth because the piecewise defined function [consisting of L1(x) for x 100 doesn€™t have a continuous second
Use mathematical induction to show that if f(x) = xex, then f (n) (x) = (x + n) ex.
If f(x) = xe sin x, find f’(x) graph f and f’(x) f’ on the same screen and comment
(a) If f’(x) = x√5 – x , find f’(x). (b) Find equations of the tangent lines to the curve y = x√5 – x at the points (1, 2) and (4, 4). (c) Illustrate part (b) by graphing the
(a)If f’(x) = 4x – tan x, – π/2 < x < π/2, find f’ and f’’. (b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f, f’, and f’’
If f(x) = (x – a) (x – b) (x – c), show that f’(x)/ f(x) = 1/x – a + 1/x – b + 1/x – c
Suppose that h(x) = f(x)g(x) and F(x) = f(g)(x)), where f(2) = 3, g(2) = 5, g’(2) = 4, f’(2) = – 2, and f’(5) = 11. Find (a) h’(2) and (b) F’(x)
If f and g are the functions whose graphs are shown, let P(x) = f(x)g(x), Q(x) = f(x)/g(x), and C(x) = f(g(x)). Find(a) P€™ (2),(b) Q€™ (2) and(c) C€™ (2)
(a) Graph the function f(x) = x – 2 sin x in the viewing rectangle [0, 8] by [– 2, 8].(b) On which interval is the average rate of change larger: [1, 2] or [2, 3]?(c) At which value of is the
(a) Find an equation of the tangent to the curve y = ex that is parallel to the line x – 4y = 1.(b) Find an equation of the tangent to the curve y = ex that passes through the origin.
An equation of motion of the form s = Ae-ct cos (wt + δ) represents damped oscillation of an object. Find the velocity and acceleration of the object.
The cost, in dollars, of producing units of a certain commodity is C(x) = 920 + 2x - 0.02x2 + 0.0007x3(a) Find the marginal cost function.(b) Find C'(100) and explain its meaning.(c) Compare C'(100)
A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2cm3/s, how fast is the water level rising when the water is 5 cm deep?
A balloon is rising at a constant speed of 5ft/s. A boy is cycling along a straight road at a speed of 15ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance
A water skier skis over the ramp shown in the figure at a speed of 30ft/s. How fast is she rising as she leaves the ramp?
The angle of elevation of the Sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the Sun is π/6?
(a) Find the linear approximation to f(x) = √25 – x2 near 3. (b) Illustrate part (a) by graphing f and the linear approximation. (c) For what values of is the linear approximation accurate
(a) Find the linearization of f(x) = 3√1 + 3x at a = 0. State the corresponding linear approximation and use it to give an approximate value for 3√1.3. (b) Determine the values of for
A window has the shape of a square surmounted by a semicircle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate
Suppose f is a differentiable function such that f(g(x)) = x and f’(x) = 1 + [f(x)]2. Show that g’(x) = 1/(1 + x2).
Show that the length of the portion of any tangent line to the asteroid x2/3 + y2/3 = a2/3 cut off by the coordinate axes is constant.
State each of the following differentiation rules both in symbols and in words.(a) The Power Rule(b) The Constant Multiple Rule(c) The Sum Rule(d) The Difference Rule(e) The Product Rule(f) The
(a) How is the number defined?(b) Express as a limit.(c) Why is the natural exponential function y = ex used more often in calculus than the other exponential functions y = ax?(d) Why is the natural
(a) Explain how implicit differentiation works.(b) Explain how logarithmic differentiation works.
What are the second and third derivatives of a function f? If f is the position function of an object, how can you interpret and f’’’?
(a) Write an expression for the linearization of f at a. (b) If y = f(x), write an expression for the differential dy. (c) If dx = ∆x, draw a picture showing the geometric meanings of
Find points P and Q on the parabola y = 1 €“ x2 so that the triangle ABC formed by the-axis and the tangent lines at P and Q is an equilateral triangle.
Find the point where the curves y = x3 – 3x + 4 and y = x(x2 – x) are tangent to each other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent.
Show that sin–1 (tanh x) = tan–1 (sinh x).
A car is traveling at night along a highway shaped like a parabola with its vertex at the origin (see the figure). The car starts at a point 100 m west and 100 m north of the origin and travels in an
A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the
Find the nth derivative of the function f(x) = xn / (1 – x).
The figure shows a circle with radius 1 inscribed in the parabola y = x2. Find the center of the circle.
If f is differentiable at a, where a > 0, evaluate the following limit in terms of f€™ (a):
The figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length 1.2 m. The pin P slides back and forth along the -axis as the wheel rotates counterclockwise at a rate of 360
Tangent lines T1 and T2 are drawn at two point€™s p1 and p2 on the parabola y = x2 and they intersect at a point P. Another tangent line T is drawn at a point between P1 and P2; it
Let T and N be the tangent and normal lines to the ellipse x2/9 + y2/4 = 1 at any point P on the ellipse in the first quadrant. Let xT and yT be the - and -intercepts of T and xN and yN be the
(a) Use the identity for tan (x – y) (see Equation 14b in Appendix D) to show that if two lines L1 and L2 intersect at an angle a, then where m1 and m2 are the slopes of L1 and L2, respectively.(b)
Let P(x1, y1) be a point on the parabola y2 = 4px with focus F(p, 0). Let a be the angle between the parabola and the line segment FP, and let β be the angle between the horizontal line and the
Suppose that we replace the parabolic mirror of Problem 16 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle
(a) The cubic function f(x) = x(x – 2)(x – 6) has three distinct zeros: 0, 2, and 6. Graph f and its tangent lines at the average of each pair of zeros. What do you notice?(b) Suppose the cubic

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