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study help
mathematics
calculus 6th edition
Calculus 6th Edition James Stewart - Solutions
Prove that d dx (sinh x) = 1 √1 + x²
Differentiate:(a) f(x) = 1/x2(b) y = 3√x2
Find the 27th derivative of cos x.
If x, y, and z are positive numbers, prove that (x² + 1)(y² + 1)(z² + 1) xyz 8
Solved/dx(x8 + 12x5 – 4x4 + 10x3 – 6x + 5)
A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area?
Graph the polynomial f(x) = 2x6 + 3x5 + 3x3 – 2x2. Use the graphs of f' and f" to estimate all maximum and minimum points and intervals of concavity.
Starting with x1 = 2, find the third approximation x3 to the root of the equation x3 – 2x –5 = 0.
Find all functions g such that g'(x) = 4 sin x + 2.x² -√√x X
Find the most general antiderivative of each of the following functions. (a) f(x) = sin x (b) f(x) = 1/x (c) f(x) = xn, n ≠ –1
Draw the graph of the functionin a viewing rectangle that contains all the important features of the function. Estimate the maximum and minimum values and the intervals of concavity. Then use calculus to find these quantities exactly. f(x) = = x² + 7x + 3 z.X² et
A cylindrical can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can.
Use Newton's method to find 6√2 correct to eight decimal places.
Find f if f'(x) = ex + 20(1 + x2)–1 and f(0) = –2.
Graph the function f(x) x²(x + 1)³ (x - 2)²(x-4)4*
Find the point on the parabola y2 = 2x that is closest to the point (1, 4).
A man launches his boat from point A on a bank of a straight river, 3 km wide, and wants to reach point B, 8 km downstream on the opposite bank, as quickly as possible (see Figure 7). He could row his boat directly across the river to point C and then run to B, or he could row directly to B, or he
Find f if f"(x) = 12x2 + 6x – 4, f(0) = 4, and f(1) = 1.
Graph the function f(x) = sin(x + sin 2x). For 0 ≤ x ≤ π , estimate all maximum and minimum values, intervals of increase and decrease, and inflection points correct to one decimal place.
The graph of a function f is given in Figure 3. Make a rough sketch of an antiderivative F, given that F(0) = 2.
Find the area of the largest rectangle that can be inscribed in a semicircle of radius r.
How does the graph of f(x) = 1/(x2 + 2x + c) vary as c varies?
A particle moves in a straight line and has acceleration given by a(t) = 6t + 4. Its initial velocity is v(0) = –6 cm/s and its initial displacement is s(0) = 9 cm. Find its position function s(t).
A store has been selling 200 DVD burners a week at $350 each. A market survey indicates that for each $10 rebate offered to buyers, the number of units sold will increase by 20 a week. Find the demand function and the revenue function. How large a rebate should the store offer to maximize its
Find the general indefinite integral∫(10x4 – 2 sec2x) dx
Use rectangles to estimate the area under the parabola y = x2 from 0 to 1 (the parabolic region S illustrated in Figure 3). y 0 y=x² S 1 AX x
(a) Prove that if f is a continuous function, then(b) Use part (a) to show thatfor all positive numbers . fª ƒ(x) dx = f*ª ƒ (a − x) dx
Evaluate √9 - x² dx. -2 x-²
Use (a) The Trapezoidal Rule (b) The Midpoint Rule with n = 5 to approximate the integral ∫12(1/x) dx.
Evaluate So%² xex dx. -00
Evaluate x² + 2x - 1 - dx. 2x³ + 3x²2x
Evaluate ∫In x dx.
Use the Table of Integrals to find x² √√5 - 4x² dx.
Find ∫sin5x cos2x dx.
Find the area enclosed by the ellipse 2.-1². a² + b² = 1
Find 1 x² √√x² + 4 dx.
Evaluate 1 J-00 1 + x² 00 dx.
Find where a ≠ 0. TD - zr X dx
Find ∫t2et dt.
Evaluate ∫0π sin2x dx.
Find S X √x² + 4 dx.
(a) Use the Midpoint Rule with n = 10 to approximate the integral ∫01ex2 dx. (b) Give an upper bound for the error involved in this approximation.
For what values of p is the integralconvergent? 1 XP xp.
Use the Table of Integrals to find S x√x² + 2x + 4 dx.
Find x42x² + 4x + 1 x³x²-x+1 -dx.
Evaluate ∫ex sin x dx.
Evaluatewhere a > 0. dx √x² - a² 2
Find ∫sin4x dx.
Use a computer algebra system to find fx√x² + 2x + 4 dx.
Find 12 1 x-2 -dx.
Evaluate 2x²x + 4 x³ + 4x .3 dx.
Evaluate ∫tan6x sec4x dx.
Evaluate X /3 - 2x - x² d.x.
Use a CAS to evaluate ∫x(x2 + 5)8 dx.
Use a CAS to find ∫sin5x cos2x dx.
The arc of the parabola y = x2 from (1, 1) to (2, 4) is rotated about the y-axis. Find the area of the resulting surface.
Find the hydrostatic force on one end of a cylindrical drum with radius 3 ft if the drum is submerged in water 10 ft deep.
A 5-mg bolus of dye is injected into a right atrium. The concentration of the dye (in milligrams per liter) is measured in the aorta at one-second intervals as shown in the chart. Estimate the cardiac output.
Find the mean of the exponential distribution of Example 2:Data from Example 2Phenomena such as waiting times and equipment failure times are commonly modeled by exponentially decreasing probability density functions. Find the exact form of such a function. So f(t) Ice-cr if t < 0 if t = 0 <
Phenomena such as waiting times and equipment failure times are commonly modeled by exponentially decreasing probability density functions. Find the exact form of such a function.
(a) Set up an integral for the length of the arc of the hyperbola xy = 1 from the point (1, 1) to the point (2,1/2). (b) Use Simpson's Rule with n = 10 to estimate the arc length.
Find the moments and center of mass of the system of objects that have masses 3, 4, and 8 at the points (–1, 1), (2, –1), and (3, 2), respectively.
Find the arc length function for the curve y = x2 – 1/8In x taking P0(1, 1) as the starting point.
Find the center of mass of a semicircular plate of radius r.
Suppose the average waiting time for a customer’s call to be answered by a company representative is five minutes.(a) Find the probability that a call is answered during the first minute.(b) Find the probability that a customer waits more than five minutes to be answered.
Find the centroid of the region bounded by the curves y = cos x, y = 0, x = 0, and x = π/2.
If the ellipse of Example 2 is rotated through an angle π/4 about the origin, find a polar equation and graph the resulting ellipse.Data from Example 2Find a polar equation for a parabola that has its focus at the origin and whose directrix is the line y = –6.
Find the length of one arch of the cycloid x = r(θ – sin θ), y = r(1 – cos θ).
(a) Solve the differential equation dy/dx = x2/y2. (b) Find the solution of this equation that satisfies the initial condition y(0) = 2.
Draw a direction field for the logistic equation with k = 0.08 and carrying capacity K = 1000. What can you deduce about the solutions?
Solve the differential equation dy/dx + 3x2y = 6x2.
Find a solution of the differential equation y' = 1/2(y2 – 1) that satisfies the initial condition y(0) = 2.
Suppose that in the simple circuit of Figure 9 the resistance is 12Ω, the inductance is 4 H, and a battery gives a constant voltage of 60 V.(a) Draw a direction field for Equation 1 with these values. (b) What can you say about the limiting value of the current? (c) Identify any equilibrium
Write the solution of the initial-value problemand use it to find the population sizes P(40) and P(80). At what time does the population reach 900? dP dt 0.08P 1 P 1000 P(0) = 100
Find the solution of the initial-value problem x²y + xy = 1 x > 0 y(1) = 2
Use Euler’s method with step size 0.1 to construct a table of approximate values for the solution of the initial-value problem y' = x + y y(0) = 1
Solve the equation y' = x2y.
Find the exponential and logistic models for Gause’s data. Compare the predicted values with the observed values and comment on the fit.
In Section 9.2 we modeled the current I(t) in the electric circuit shown in Figure 5 by the differential equationFind an expression for the current in a circuit where the resistance is 12Ω, the inductance is 4 H, a battery gives a constant voltage of 60 V, and the switch is turned on when t = 0.
In Example 2 we discussed a simple electric circuit with resistance 12Ω, inductance 4 H, and a battery with voltage 60 V. If the switch is closed when t = 0, we modeled the current I at time t by the initial-value problemEstimate the current in the circuit half a second after the switch is
Solve y' + 2xy = 1.
Suppose that in the simple circuit of Figure 4 the resistance is 12Ω and the inductance is 4 H. If a battery gives a constant voltage of 60 V and the switch is closed when t = 0 so the current starts with I(0) = 0, find (a) I(t), (b) The current after 1 s, (c) The limiting value of the current.
Find the orthogonal trajectories of the family of curves x = ky2, where k is an arbitrary constant.
Suppose that the resistance and inductance remain as in Example 4 but, instead of the battery, we use a generator that produces a variable voltage of E(t) = 60 sin 30t volts. Find I(t).
A tank contains 20 kg of salt dissolved in 5000 L of water. Brine that contains 0.03 kg of salt per liter of water enters the tank at a rate of 25 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt remains in the tank after half an hour?
A curve C is defined by the parametric equations x = t2, y = t3 - 3t. (a) Show that C has two tangents at the point (3, 0) and find their equations.(b) Find the points on C where the tangent is horizontal or vertical. (c) Determine where the curve is concave upward or downward. (d) Sketch the
Find the area enclosed by one loop of the four-leaved rose r = cos 2θ.
Sketch and identify the curve defined by the parametric equations x = t² - 2t y = 1 + 1
Find the focus and directrix of the parabola y2 + 10x = 0 and sketch the graph.
Find a polar equation for a parabola that has its focus at the origin and whose directrix is the line y = –6.
(a) Find the tangent to the cycloid x = r(θ – sin θ), y = r(1 – cosθ) at the point where θ = π/3. (b) At what points is the tangent horizontal? When is it vertical?
Convert the point (2, π/3) from polar to Cartesian coordinates.
Find the area of the region that lies inside the circle r = 0. 3 sin θ and outside the cardioid = 1 + sinθ.
Sketch the graph of 9x2 + 16y2 = 144 and locate the foci.
What curve is represented by the following parametric equations? x = cos t y = sin t 0 ≤ t ≤2TT
A conic is given by the polar equationFind the eccentricity, identify the conic, locate the directrix, and sketch the conic. 1 = 10 3-2 cos 0
Find the area under one arch of the cycloid x = r (θ – sin θ) y = (1 – cos θ) (See Figure 3.)Figure 3 y 0 2πr X
Represent the point with Cartesian coordinates (1, –1) in terms of polar coordinates.
Find all points of intersection of the curves = cos 2θ and r = 1/2.
What curve is represented by the given parametric equations? x = sin 21 2t y = cos 2t 0 ≤ t ≤2TT
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