Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface.
> Use the guidelines in Section 4.5 to sketch the curve y = x sin x, 0 ≤ x ≤ 2 π. Use Newton’s method when necessary.
> Use Newton’s method to find the absolute maximum value of the function f (t) = cos t + t - t2 correct to eight decimal places.
> Use Newton’s method to find all solutions of the equation sin x = x2 - 3x + 1 correct to six decimal places.
> Use Newton’s method to find the root of the equation x5 - x4 + 3x2 - 3x - 2 = 0 in the interval [1, 2] correct to six decimal places.
> A manufacturer determines that the cost of making x units of a commodity is C(x) = 1800 + 25x - 0.2x2 + 0.001x3 and the demand function is p(x) = 48.2 - 0.03x. (a) Graph the cost and revenue functions and use the graphs to estimate the production level f
> A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at $12, average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average attendance w
> The figure shows the graph of the derivative f ‘ of a function f. (a) On what intervals is f increasing or decreasing? (b) For what values of x does f have a local maximum or minimum? (c) Sketch the graph of f ’â
> A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a
> The mass of the part of a metal rod that lies between its left end and a point x meters to the right is 3x2 kg. Find the linear density (see Example 2) when x is (a) 1 m, (b) 2 m, and (c) 3 m. Where is the density the highest? The lowest?
> Solve Exercise 55 when |CD| = 2 cm. Exercise 55: In ΔABC, D lies on AB, CD AB, |AD| = |BD| = 4 cm, and |CD| = 5 cm. Where should a point P be chosen on CD so that the sum |PA | + |PB | + |PC | is a minimum?
> In ΔABC, D lies on AB, CD AB, |AD| = |BD| = 4 cm, and |CD| = 5 cm. Where should a point P be chosen on CD so that the sum |PA | + |PB | + |PC | is a minimum?
> Find the volume of the largest circular cone that can be inscribed in a sphere of radius r.
> Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r.
> Find the point on the hyperbola xy = 8 that is closest to the point (3, 0).
> 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.
> Sketch the graph of a function that satisfies the given conditions. f is odd, f ‘(x) f'(x) > 0 for x > 2, f"(x) > 0 for 0 < x< 3, f"(x) < 0 for x > 3, lim f(x) = -2 X 00
> Let g(x) = f (x2), 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 does g have an extreme value? (b) Discuss the concavity of g.
> For what values of the constants a and b is (1, 3) a point of inflection of the curve y = ax3 + bx2?
> By applying the Mean Value Theorem to the function f (x) = x1/5 on the interval [32, 33], show that 2 < 5 33 < 2.0125
> (a) The volume of a growing spherical cell is V = 4/3 πr3, where the radius r is measured in micrometers (1 μm − 1026 m). Find the average rate of change of V with respect to r when r changes from (i) 5 to 8 μm (ii) 5 to 6 μm (iii) 5 to 5.1 μm (b) Find t
> 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.
> Show that the equation 3x + 2 cos x + 5 = 0 has exactly one real root.
> Investigate the family of functions f (x) = ln(sin x + C). What features do the members of this family have in common? How do they differ? For which values of C is f continuous on (-∞, ∞)? For which values of C does f have no graph at all? What happens a
> (a) Find the average rate of change of the area of a circle with respect to its radius r as r 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 = 2. (c) Show that the rate of change of the area of a cir
> (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 x approaches ∞, -∞, 0+, and 0-. (c) Use the graph of f to estimate the coordinates of the inflection points. (d) Use your CAS to compute a
> 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 concavity, and inflection points. In Exercise 35 use calculus to find thes
> 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 concavity, and inflection points. In Exercise 35 use calculus to find thes
> A spherical balloon is being inflated. Find the rate of increase of the surface area (S = 4πr2) with respect to the radius r when r is (a) 1 ft, (b) 2 ft, and (c) 3 ft. What conclusion can you make?
> Use the guidelines of Section 4.5 to sketch the curve. y = x + ln(x2 + 1)
> Use the guidelines of Section 4.5 to sketch the curve. y = (x – 2)e-x
> Sketch the graph of a function that satisfies the given conditions. f (0) = 0, f is continuous and even, f ‘(x) = 2x if 0 < x < 1, f ‘(x) = -1 if 1 < x < 3, f ‘(x) = 1 if x > 3
> Use the guidelines of Section 4.5 to sketch the curve. y = sin-1(1/x)
> Use the guidelines of Section 4.5 to sketch the curve. y = 4x - tan x, -π/2 < x
> Use the guidelines of Section 4.5 to sketch the curve. y = ex sin x, -π ≤ x ≤ π
> 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 s. What can you conclude?
> Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is a little more than 12
> Differentiate. y = c cos t + t2 sin t
> (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, calculate dV / dx when x = 3 mm and explain its meaning. (b
> Find the local and absolute extreme values of the function on the given interval. f (x) = x2e-x, [-1, 3]
> Find the local and absolute extreme values of the function on the given interval. f (x) = x + 2 cos x, [-π, π ]
> Traditionally, the earth’s surface has been modeled as a sphere, but the World Geodetic System of 1984 (WGS-84) uses an ellipsoid as a more accurate model. It places the center of the earth at the origin and the north pole on the positive z-axis. The dis
> Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same radius r. The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the bo
> A solid has the following properties. When illuminated by rays parallel to the z-axis, its shadow is a circular disk. If the rays are parallel to the y-axis, its shadow is a square. If the rays are parallel to the x-axis, its shadow is an isosceles trian
> Find an equation of the largest sphere that passes through the point (-1, 1, 4) and is such that each of the points (x, y, z) inside the sphere satisfies the condition x2 + y2 + z2, 136 + 2 (x + 2y + 3z)
> A plane is capable of flying at a speed of 180 km/h in still air. The pilot takes off from an airfield and heads due north according to the plane’s compass. After 30 minutes of flight time, the pilot notices that, due to the wind, the plane has actually
> Let L be the line of intersection of the planes cx + y + z = c and x - cy + cz = 21, where c is a real number. (a). Find symmetric equations for L. (b). As the number c varies, the line L sweeps out a surface S. Find an equation for the curve of intersec
> Let B be a solid box with length L, width W, and height H. Let S be the set of all points that are a distance at most 1 from some point of B. Express the volume of S in terms of L, W, and H.
> Differentiate. g(θ ) = e θ (tan θ - θ )
> Find dy/dx by implicit differentiation. x3 - xy2 + y3 = 1
> (a) Graph several members of the family of functions f (x) = (2cx - x2)/c3 for c > 0 and look at the regions enclosed by these curves and the x-axis. Make a conjecture about how the areas of these regions are related. (b) Prove your conjecture in part (a
> Find the minimum value of the area of the region under the curve y = x + 1/x from x = a to x = a + 1.5, for all a > 0.
> The figure shows a region consisting of all points inside a square that are closer to the center than to the sides of the square. Find the area of the region. 2 2 2. 2.
> Given the point (a,b) in the first quadrant, find the downward-opening parabola that passes through the point (a,b) and the origin such that the area under the parabola is a minimum.
> The figure shows a parabolic segment, that is, a portion of a parabola cut off by a chord AB. It also shows a point C on the parabola with the property that the tangent line at C is parallel to the chord AB. Archimedes proved that the area of the parabol
> Find an equation for the surface consisting of all points that are equidistant from the point s21, 0, 0d and the plane x = 1. Identify the surface.
> (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 side length x changes. Find A’(15) and explain its meani
> Find an equation for the surface obtained by rotating the line z = 2y about the z-axis.
> Find the point on the parabola y = 1 - x2 at which the tangent line cuts from the first quadrant the triangle with the smallest area.
> A particle moves with position function s = t4 - 4t3 - 20t2 + 20t t ≥ 0 (a) At what time does the particle have a velocity of 20 m/s? (b) At what time is the acceleration 0? What is the significance of this value of t?
> Show that the inflection points of the curve y = (sin x)/x lie on the curve y2 (x4 + 4) = 4.
> Show that x2y2 (4 - x2)(4 - y2) ≤ 16 for all numbers x and y such that |x | ≤ 2 and |y | ≤ 2.
> Does the function f (x) = e10| x-2| - x2 have an absolute maximum? If so, find it. What about an absolute minimum?
> Show that |sin x - cos x | ≤ 2 for all x.
> Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely?
> Given a sphere with radius r, find the height of a pyramid of minimum volume whose base is a square and whose base and triangular faces are all tangent to the sphere. What if the base of the pyramid is a regular n-gon? (A regular n-gon is a polygon with
> One of the problems posed by the Marquis de l’Hospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceil
> For what values of c is there a straight line that intersects the curve y = x4 + cx3 + 12x2 - 5x + 2 in four distinct points?
> If a rock is thrown vertically upward from the surface of Mars with velocity 15 m/s, its height after t seconds is h = 15t - 1.86t2. (a) What is the velocity of the rock after 2 s? (b) What is the velocity of the rock when its height is 25 m on its way u
> The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness h of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is greater than the speed in the upper layer. A dynamite cha
> Let f(x) = a1 sin x + a2 sin 2x + ∙ ∙ ∙ + an sin nx, where a1, a2, . . . , an are real numbers and n is a positive integer. If it is given that | f (x) | ≤ |sin x | for all x, show that |a1 + 2a2 + ∙ ∙ ∙ + nan | ≤ 1
> For which positive numbers a does the curve y = ax intersect the line y = x?
> ABCD is a square piece of paper with sides of length 1 m. A quarter-circle is drawn from B to D with center A. The piece of paper is folded along EF, with E on AB and F on AD, so that A falls on the quarter-circle. Determine the maximum and minimum areas
> The line y = mx + b intersects the parabola y = x2 in points A and B. (See the figure.) Find the point P on the arc AOB of the parabola that maximizes the area of the triangle PAB. y y=x? B A y = mx +b
> Use a computer with three-dimensional graphing software to graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface. -4x2 - y2 + z2 = 1
> Sketch the region in the plane consisting of all points (x, y) such that 2xy ≤ |x - y | ≤ x2 + y2
> An isosceles triangle is circumscribed about the unit circle so that the equal sides meet at the point s0, ad on the y-axis (see the figure). Find the value of a that minimizes the lengths of the equal sides. (You may be surprised that the result does no
> For what values of c does the curve y = cx3 + ex have inflection points?
> If a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is s = 80t - 16t2. (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is 96 ft above the ground on its way up?
> If P(a, a2) is any point on the parabola y = x2, except for the origin, let Q be the point where the normal line at P intersects the parabola again (see the figure). (a) Show that the y-coordinate of Q is smallest when a = 1/ 2 . (b) Show that the line s
> Sketch the set of all points (x, y) such that |x + y | ≤ ex.
> Find the highest and lowest points on the curve x2 + xy + y2 = 12.
> Calculate the given quantity if a = i + j - 2k b = 3i - 2j + k c = j - 5k (a). 2a + 3b (b). |b | (c). a ∙ b (d). a × b (e). |b × c | (f). a ∙ (b × c) (g). c × c (h). a × (b × c) (i). compa b (j). proja b (k). The angle between a and b (correct to th
> Differentiate the function. f (x) = ln (1/x)
> (a). Find an equation of the sphere that passes through the point (6, -2, 3) and has center (-1, 2, 1). (b). Find the curve in which this sphere intersects the yz-plane. (c). Find the center and radius of the sphere x2 + y2 + z2 - 8x + 2y + 6z + 1 = 0
> An ellipsoid is created by rotating the ellipse 4x2 + y2 – 16 about the x-axis. Find an equation of the ellipsoid.
> (a). Find an equation of the plane that passes through the points A (2, 1, 1), B (-1, -1, 10), and C (1, 3, -4). (b). Find symmetric equations for the line through B that is perpendicular to the plane in part (a). (c). A second plane passes through (2, 0
> Find an equation of the plane through the line of intersection of the planes x - z = 1 and y + 2z = 3 and perpendicular to the plane x + y - 2z = 1.
> 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 t seconds. (a) Find the velocity after 2 s and after 4 s. (b) When does the projectile r
> Determine whether the lines given by the symmetric equations are parallel, skew, or intersecting. x - х — 1 у - 2 г-3 y z - 3 2 3 4 у — 3 y -1 х+1 z + 5
> Find the distance from the origin to the line x = 1 + t, y = 2 - t, z = 21 + 2t.
> Find the point in which the line with parametric equations x = 2 - t, y = 1 + 3t, z = 4t intersects the plane 2x - y + z = 2.
> Find an equation of the plane. The plane through (1, 2, -2) that contains the line x = 2t, y = 3 - t, z = 1 + 3t
> Find an equation of the plane. The plane through (3, -1, 1), (4, 0, 2), and (6, 3, 1)
> Reduce the equation to one of the standard forms, classify the surface, and sketch it. x2 - y2 - z2 - 4x - 2z + 3 = 0
> A boat is pulled onto shore using two ropes, as shown in the diagram. If a force of 255 N is needed, find the magnitude of the force in each rope. 20° 255 N 30°
> Find the acute angle between two diagonals of a cube.
> Show that if a, b, and c are in V3, then (a × b) ∙ [(b × c) × (c × a) g = [a ∙ (b × c)]2
> Suppose that u ∙ (v × w) = 2. Find (a). (u × v) ∙ w (b). u ∙ (w × v) (c). v ∙ (u × w) (d). (u × v) ∙ v
> (a). Sketch the graph of y = ex as a curve in R2. (b). Sketch the graph of y = ex as a surface in R3. (c). Describe and sketch the surface z = ey.
> Differentiate the function. y = 1 / ln x
