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mathematics
calculus

Questions and Answers of Calculus

If F(x) = f(3f(4f (x))), where f(0) = 0 and f'(0) = 2, find F'(0).
Show that the function y = e2x(A cos 3x + B sin 3x) satisfies the differential equation y" - 4y' + 13y = 0.
Find the 50th derivative of y = cos 2x.
The displacement of a particle on a vibrating string is given by the equation s(t) = 10 + 1/4 sin(10 πt) where is measured in centimeters and in seconds. Find the velocity of the particle after
A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maxi - mum
The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine
A particle moves along a straight line with displacement s(t) velocity v(t), and acceleration a(t). Show that A(t) = v(t) dv/dx Explain the difference between the meanings of the derivatives dv/dt
The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off.The following data describe the charge Q remaining on the capacitor (measured
Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer. (a) Use a CAS to
Use the Chain Rule to prove the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function.
(a) If is a positive integer, prove that d/dx (sinn x cos nx) = n sinn-1 x cos(n + 1)x (b) Find a formula for the derivative of y = cosn x cos nx that is similar to the one in part (a).
Use the Chain Rule to show that if is measured in degrees, then d/dθ (sin θ) = π / 180 cos θ (This gives one reason for the convention that radian measure is always used when dealing with
If y = f(u) and u = g(x), where f and g are twice different tiable functions, show that
(a) Find y' by implicit differentiation. (b) Solve the equation explicitly for and differentiate to get y' in terms of x.
If f(x) + x2 [f(x)]3 = 10 and f(1) = 2, find f`'(1).
Regard as the independent variable and as the dependent variable and use implicit differentiation to find dx/dy. x4y2 - x3y + 2xy3 = 0
Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y sin 2x = x cos 2y, (π/2, π/4)
(a) The curve with equation y2 = 5x4 - x2 is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point (1, 2). (b) Illustrate part (a) by graphing the curve and the
Find y" by implicit differentiation. (a) 9x2 + y2 = 9 (b) x3 + y3 = 1
If xy + ey = e, find the value of y" at the point where x = 0.
Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems. (a) Graph the curve with equation Y(y2 - 1) (y - 2) = x(x - 1) (x - 2) At how many points does
Find the points on the lemniscates in Exercise 31 where the tangent is horizontal.
Find an equation of the tangent line to the hyperbola x2/a2 - y2/b2 = 1 at the point (x0, y0).
Show, using implicit differentiation, that any tangent line at a point P to a circle with center O is perpendicular to the radius OP.
Find the derivative of the function. Simplify where possible. (a) y = (tan-1 x)2 (b) y = sin-1(2x + 1) (c) G(x) = √1 - x2 arccos x
Find dy/dx by implicit differentiation. (a) x3 + y3 = 1 (b) x2 + xy - y2 = 4 (c) x4(x + y) = x2 (3x - y)
Find f'(x). Check that your answer is reasonable by comparing the graphs of f and f'/ F(x) = √1 - x2 arcsin x
Prove the formula for (d/dx) (cos-1x) by the same method as for (d/dx) (sin-1x).
Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each other; that is, every
Show that the ellipse x2/a2 + y2/b2 = 1 and the hyperbola x2 / A2 - y2/B2 = 1 are orthogonal trajectories if A2 < a2 and a2 - b2 = A2 + B2 (so the ellipse and hyperbola have the same foci).
(a) The van der Waals equation for moles of a gas iswhere P is the pressure, V is the volume, and is the temperature of the gas. The constant R is the universal gas constant and and are positive
The equation x2 - xy + y2 = 3 represents a "rotated ellipse," that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the -axis and show
Find all points on the curve x2 y2 + xy = 2 where the slope of the tangent line is -1.
(a) Suppose f is a one-to-one differentiable function and its inverse function f-1 is also differentiable. Use implicit differentiation to show thatprovided that the denominator is not 0. (b) If f(4)
The Bessel function of order 0, y = J(x), satisfies the differential equation xy" + y' + xy = 0 for all values of and its value at 0 is J(0) = 1. (a) Find J'(0). (b) Use implicit differentiation to
Find y' and y". (a) y = x2 ln(2x) (b) y = ln (x + √1 + x2)
Differentiate f and find the domain of f. (a) f(x) = x/ 1 - ln(x - 1) (b) f(x) = ln(x2 - 2x)
Differentiate the function. (a) f(x) = sin (ln x) (b) f(x) = ln 1/x (c) f(x) = log10(x3 + 1)
If f(x) = ln x/ x2, find f'(1).
Find an equation of the tangent line to the curve at the given point. y = ln(x2 - 3x + 1), (3, 0)
If f(x) = sin x + ln x, find f'(x). Check that your answer is reasonable by comparing the graphs of f and f'.
Let f(x) = cx + ln(cos x). For what value of is f'(π/4) = 6?
Use logarithmic differentiation to find the derivative of the function. y = (x2 + 2)2 (x4 + 4)4
Find y' if y = ln(x2 + y2)
Find a formula for f(n)(x) if f(x) = ln(x - 1).
Use the definition of derivative to prove that
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
(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) 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 spherical balloon is being inflated. Find the rate of increase of the surface area (S = 4πr2) with respect to the radius r when is (a) 1 ft, (b) 2 ft, (c) 3 ft. What conclusion can you make?
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 when is (a) 1 m, (b) 2 m, (c) 3 m. Where is the density the
The quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time (measured in seconds) is given by Q(t) = t3 - 2t2 + 6t + 2. Find the current when (a) t = 0.5 s (b) t =
The force F acting on a body with mass m and velocity is the rate of change of momentum: F = (d/dt)(m), If m is constant, this becomes F = ma, where a = dv/dt is the acceleration. But in the theory
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 with
In Example 6 we considered a bacteria population that doubles every hour. Suppose that another population of bacteria triples every hour and starts with 400 bacteria. Find an expression for the
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
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 velocity of
The cost, in dollars, of producing yards of a certain fabric is C(x) = 1200 + 12x - 0.1x2 + 0.0005x3 (a) Find the marginal cost function. (b) Find C'(200) and explain its meaning. What does it
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(x) = p(x) /x (a) Find A'(x). Why does the company
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 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
Graphs of the velocity functions of two particles are shown, where is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.(a)(b)
The height (in meters) of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of 24.5 m/s is h = 2 + 24.5t - 4.9t2 after seconds. (a) Find the velocity
If a rock is thrown vertically upward from the surface of Mars with velocity 15m/s, its height after seconds is h = 15t - 1.86t2. (a) What is the velocity of the rock after 2 s? (b) What is the
A population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days.
Scientists can determine the age of ancient objects by the method of radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon,
A roast turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 75°F. (a) If the temperature of the turkey is 150°F after
When a cold drink is taken from a refrigerator, its temperature is C. After 25 minutes in a C room its temperature has increased to 10°C. (a) What is the temperature of the drink after 50
The rate of change of atmospheric pressure P with respect to altitude is proportional to P, provided that the temperature is constant. At 15°C the pressure is 101.3 kPa at sea level and 87.14 kPa at
(a) If $3000 is invested at 5% interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily,
A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. (a) Find an expression for the number of bacteria
The table gives estimates of the world population, in millions, from 1750 to 2000.(a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900
Experiments show that if the chemical reaction N2O5 → 2NO2 + 1/2O2 takes place at 45°C, the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows: - d[N2O5]/dt =
The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. (a) Find the mass that remains after years. (b) How much of the sample remains after 100 years? (c) After how long will only
A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when
Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?
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 P. At what rate are the people moving apart 15 min after
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
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 PM?
Water is leaking out of an inverted conical tank at a rate of 10,000 cm/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 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
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
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
Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 cm2?
The top of a ladder slides down a vertical wall at a rate of 0.5 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. How long is
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 instant
If two resistors with resistances R1 and R2 are connected in parallel, as in the figure, then the total resistance R, measured in ohms (Î©), is given by1/R = 1/R1 + 1/R2If R1 and R2 are
Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2°/min. How fast is the length of the third side increasing when the angle between the sides of
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 plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing at a rate of π/6 rad/min.
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 30 °. At what rate is the distance from the plane to the radar
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
A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3/ min. How fast is the height of the water increasing?
Suppose y = √2x + 1, where and are functions of . (a) If dx/dt = 3, find dy/dt when x = 4. (b) If dy/dt = 5, find dx/dt when x = 12.
If x2 + y2 + z2 = 9, dx/dt = 5, and dy/dt = 4, find dz/dt when (x, y, z) = (2, 2, 1).
Calculate y'. (a) y = (x2 + x3)4 (b) y = x2 - x + 2/ √x (c) y = x2 sin π x
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 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√1.03. (b) Determine the values of for which the
A window has the shape of a square surmounted by a semi - circle. 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
Evaluate ˆš1 + tan x - ˆš1 + sin x/x3.

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