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Questions and Answers of Calculus

Prove that a cubic function has exactly one point of inflection.
Prove that, if f'(x) exists and is continuous on an interval l and if f'(x) ≠ 0 at all interior points of l, then either f is increasing throughout l or decreasing throughout l.
Suppose that f is a function whose derivative is f'(x) = (x2 - x + 1) / (x2 + 1). Use Problem 37 to prove that f is increasing everywhere.
Use the Monotonicity Theorem to prove each statement if 0 < x < y. (a) x2 < y2 (b) √x < √y (c) 1/x > 1/y
What condition on a, b, and c will make f(x) = ax3 + bx2 + cx + d always increasing?
Determine a and b so that f(x) = a√x + b / √x has the point (4, 13) as an inflection point.
Suppose that the cubic function f(x) has three real zeros r1, r2, and r3, Show that its inflection point has x-coordinate (r1 + r2 + r3) / 3.
Suppose that f'(x) > 0 and g'(x) > 0 for all x. What simple additional conditions (if any) are needed to guarantee that: (a) f(x) + g(x) is increasing for all x; (b) f(x) ∙ g(x) is increasing for
Suppose that f"(x) > 0 and g"(x) > 0 for all x. What simple additional conditions (if any) are needed to guarantee that (a) f(x) + g(x) is concave up for all x; (b) f(x) ∙ g(x) is concave up for
Let f(x) = sin x + cos(x/2) on the interval l = (- 2, 7). (a) Draw the graph off on l. (b) Use this graph to estimate where (x) < 0 on I. (c) Use this graph to estimate where f"(x) < 0 on l. (d) Plot
Repeat Problem 45 for f(x) = x cos2(x/3) on (0,10). Let f(x) = sin x + cos(x/2) on the interval l = (- 2, 7). (a) Draw the graph off on l. (b) Use this graph to estimate where (x) < 0 on I. (c) Use
Translate each of the following into the language of derivatives of distance with respect to time. For each part, sketch a plot of the car's position s against time t, and indicate the concavity. (a)
Translate each of the following into the language of derivatives, sketch a plot of the appropriate function and indicate the concavity. (a) Water is evaporating from the tank at a constant rate. (b)
Translate each of the following statements into mathematical language, sketch a plot of the appropriate function, and indicate the concavity. (a) The cost of a car continues to increase and at a
Translate each statement from the following newspaper column into a statement about derivatives. (a) In the United States, the ratio R of government debt to national income remained unchanged at
Coffee is poured into the cup shown in Figure 20 at the rate of 2 cubic inches per second. The top diameter is 3.5 inches, the bottom diameter is 3 inches, and the height of the cup is 5 inches. This
Water is being pumped into a cylindrical tank at a constant rate of 5 gallons per minute, as shown in Figure 21.The tank has diameter 3 feet and length 9.5 feet. The volume of the tank is
A liquid is poured into the container shown in Figure 22 at the rate of 3 cubic inches per second. The container holds about 24 cubic inches. Sketch a graph of the height h of the liquid as a
A 20-gallon barrel, as shown in Figure 23, leaks at the constant rate of 0.1 gallon per day. Sketch a plot of the height h of the water as a function of timer, assuming that the barrel is full at
What are you able to deduce about the shape of a vase based on each of the following tables, which give measurements of the volume of the water as a function of the depth.(a)(b)
In Problems 1-2, identify the critical points. Then use (a) the First Derivative Test and ( if possible) (b) the second Derivative Test to decide which of the critical points give a local maximum and
In Problem 1-2, find the critical points and use the test of your choice to decide which critical points give a local maximum value and which give a local minimum value. What are these local maximum
In Problems 1-2, find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. 1. f(x) = sin2 2x on [0, 2] 2. f(x) = 2x / x2 + 4 on [0, ∞)
In problem 1-3, the first derivative f' is given, Find all values of x that make the function f(a) a local minimum and (b) a local maximum. 1. f'(x) = x3(1 - x)2 2. f'(x) = - (x - 1) (x - 2) (x - 3)
In Problems 1-3, sketch a graph of a function with the given properties. If it is impossible to graph such a function, then indicate this and justify your answer. 1. f is differentiable, has domain
Consider f(x) = Ax2 + Bx + C with A > 0. Show that f(x) ≥ 0 for all x if and only if B2 - 4AC ≤ 0.
Consider f(x) = Ax3 + Bx2 + Cx + D with A > 0. Show that f has one local maximum and one local minimum if and only if B2 - 3AC > 0.
What conclusions can you draw about f from the information that f'(c) = f"(c) = 0 and f"'(c) > 0?
Find two numbers whose product is - 16 and the sum of whose squares is a minimum.
A farmer has 80 feet of fence with which he plans to en-close a rectangular pen along one side of his 100-foot barn, as shown in Figure 18 (the side along the barn needs no fence). What arc the
The farmer of Problem 10 decided to make three identical pens with his 80 feet of fence, as shown in Figure 19. What dimensions for the total enclosure make the area of the pens as large as possible?
Suppose that the farmer of Problems 10 has 180 feet of fence and wants the pen to adjoin to the whole side of the 100-foot bara, as shown in Figure 20. What should the dimensions be for maximum area?
A farmer wishes to fence off two identical adjoining rectangular pens, each with 900 square feet of area, as shown in Figure 21. What are x and y so that the least amount of fence is required?
A farmer wishes to fence off three identical adjoining rectangular pens (see Figure 22), each with 300 square feet of area. What should he width and length of each pen be so that the least amount of
Suppose that the outer boundary of the pens in Problem 14 requires heavy that costs $3 per foot, but that the two interval partitions require fence costing only $2 per foot. What dimensions x and y
Solve Problem 14 assuming that the area of each pen is 900 square feet. Study the solution to this problem and to Problem 14 and make a conjecture about the ratio of x/y in all problems of this type.
Find the points P and Q on the curve y = x2/4, 0 ≤ x ≤ 2√6, that are closest to and farthest from the point (0, 4).
A right circular cone is to be inscribed in another right circular cone of given volume, with the same axis and with the vertex of the inner cone touching the base of the outer cone. What must be the
A small island is 2 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 3 miles per hour and can walk 4 miles per hour, where should the
For what number does the principal square root exceed eight times the number by the largest amount?
In Problem 19, suppose that the woman will be picked up by a car that will average 50 miles per hour when she gets to the shore. Then where should she land?
In Problem 19, suppose that the woman uses a motorboat that goes 20 miles per hour. Then where should she land?
A powerhouse is located on one bank of a straight river that is w feet wide. A factory is situated on the opposite bank of the river, L feet downstream from the point A directly opposite the
At 7:00 A.M. one ship was 60 miles due east from a second ship. If the first ship sailed west at 20 miles per hour and the second ship sailed southeast at 30 miles per hour, when were they dosest
Find the equation of the line that is tangent to the ellipse b2x2 + a2y2 = a2b2 in the first quadrant and forms with the coordinate axes the triangle with smallest possible area (a and b arc positive
Find the greatest volume that a right circular cylinder can have if it is inscribed in a sphere of radius r.
Show that the rectangle with maximum perimeter that can be inscribed in a circle is a square.
What are the dimensions of the right circular cylinder with greatest curved surface area that can be inscribed in a sphere of radius r?
The illumination at a point is inversely proportional to the square of the distance of the point from the light source and directly proportional to the intensity of the light source. If two light
A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, and the other is bent to form an equilateral triangle. Where should the cut be made if (a) The sum of the two
For what number does the principal fourth root exceed twice the number by the largest amount?
A closed box in the form of a rectangular parallelepiped with a square base is to have a given volume. If the material used in the bottom costs 20% more per square inch than the material in the
An observatory is to be in the form of a right circular cylinder surmounted by a hemispherical dome. If the hemispherical dome costs twice as much per square foot as the cylindrical wall, what are
A weight connected to a spring moves along the x-axis so that its x-coordinate at time t is x = sin 2t + √3 cos 2t What is the farthest that the weight gets from the origin?
A flower bed will be in the shape of a sector of a circle (a pie-shaped region) of radius r and vertex angle θ, Find r and θ if its area is a constant A and the perimeter is a minimum.
A fence it feet high runs parallel to a tall building and is, feet from it (Figure 23). Find the length of the shortest ladder that will reach from the ground across the top of the fence to the wall
A rectangle has two corners on the x-axis and the other two on the parabola y = 12 - x2, with y ‰¥ 0 (figure 24). What are the dimensions of the rectangle of this type with maximum area?
A rectangle is to be inscribed in a semicircle of radius r, as shown in Figure 25. What are the dimensions of the rectangle if its area is. to be maximized?
Of all right circular cylinders with a given surface area, find the one with the maximum volume. The ends of the cylinders are closed.
Find the dimensions of the rectangle of greatest area that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.
Of all rectangles with a given diagonal, find the one with the maximum area.
Find two numbers whose product is -12 and the sum of whose squares is a minimum.
A humidifier uses a rotating disk of radius r, which is partially submerged in water. The most evaporation occurs when the exposed wetted region (shown as the upper shaded region in Figure 26) is
A metal rain gutter is to have 3-inch sides and a 3-inch horizontal bottom, the sides making an equal angle 0 with the bottom (Figure 27). What should 0 be in order to maximize the carrying capacity
Find the points on the parabola y = x2 that are closest to the point (0, 5).
Find the points on the parabola x = 2y2 that are closest to the point (10, 0).
What number exceeds its square by the maximum amount? Begin by convincing yourself that this number is on the interval [0, 1].
Show that for a rectangle of given perimeter K the one with maximum area is a square.
Find the volume of the largest open box that can be made from a piece of cardboard 24 inches square by cutting equal squares from the corners and turning up the sides (see Example 1).
In Problems a-c, use the Bisection Method to approximate the real root of the given equation on the given interval. Each answer should be accurate to two decimal places. a. x3 + 2x - 6 = 0; [1, 2] b.
Use Newton's Method to calculate 3√6 to five decimal places.
Use Newton's Method to calculate 4√47 to fie decimal places.
In Problems a to b, approximate the values of x that give maximum and minimum values of the function on the indicated intervals.a. f(x) = x4 + x3 + x2 + x; [-1, 1]b. f(x) = x3 + 1 / x4 + 1; [-4, 4]
Kepler's equation x = m + E sin x is important in astronomy. Use the Fixed point Algorithm to solve this equation for x when m = 0.8 and E = 0.2.
Sketch the graph of y = y = x1/3. Obviously, its only x-intercept is zero. Convince yourself that Newton's Method fails to converge. Explain this failure.
In installment buying, one would like to figure out the real interest rate (effective rate), but unfortunately this involves solving a complicated equation. If one buys an item worth $P today and
In applying Newton's Method to solve f(x) = 0, one can usually tell by simply looking at the numbers xl, r2, x3, ..... whether the sequence is converging. But even if it converges, say to I, can we
In Problems a-b, we the Fixed-Point Algorithm with xi as indicated to solve the equations to five decimal placesa. x =3/2cos x; x1 = 1b. x = 2 - sin x; x1 = 2
Consider the equation x = 2(x - x2) = g(x).a. Sketch the graph of y = x and y = g(x) using the same co-ordinate system, and thereby approximately locate the positive root of x = g(x).b. Try solving
Follow the directions of Problem 29 for x = 5(x - x2) = g(x).In Problem 29a. Sketch the graph of y = x and y = g(x) using the same co-ordinate system, and thereby approximately locate the positive
Consider x = ˆš1 + x.a. Apply the Fixed-Point Algorithm starting with x1 - 0 to find x2, x3, x4, and x5.b. Algebraically solve for x in x = ˆš 1 + x.c. Evaluate
Consider x = ˆš5 + x.a. Apply the Fixed-Point Algorithm starting with xl = 0 to find x2, x3, x4, and x5.b. Algebraically solve for x in x = ˆš5 + x.c. Evaluate
Consider x = 1 + 1/x.a. Apply the Fixed-Point Algorithm starting with x1 = 1 to find x2, x3, x4, and x5.b. Algebraically solve for x in x = 1 + 1/x.c. Evaluate the following expression. (An
Consider the equation x = x - f(x) /f'(x) and sup-pose that f'(x) ≠ 0 in an interval [a, b]. a. Show that if r is in [a, b] then r is a root of the equation x = x - f(x) /f'(x) if and only if f (r)
Experiment with the algorithm xn+1 = 2xn - ax2n Using several different values of a. a. Make a conjecture about what this algorithm computes. b. Prove your conjecture.
A rectangle has two corners on the x-axis and the other two on the curve y = cos x, with -π/2 < x < π/2. What are the dimensions of the rectangle of this type with maximum area?
Two hallways meet in a right angle as shown in Figure 6 of Section 3.4, except the widths of the hallways are 8.6 feet and 6.2 feet. What is the length of the longest thin rod that can be carried
An 8-foot-wide hallway makes a turn as shown in Figure 9. What is the length of the longest thin rod that can be carried around the corner?
An object thrown the edge of a 42-foot cliff follows the path given by y = 2x2 / 25 + x + 42 Figure 10. An observer stands 3 feet from the bottom of the cliff.a. Find the position of the object when
In Problems a-b, use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. a. The largest root of x3 + 6x2 + 9x + 1 =
Find the general antiderivative F(x) + C each of the following. a. f(x) = 5 b. f(x) = x - 4 c. f(x) = x2 + π d. f(x) = 3x2 + √3
In Problems a-c, evaluate the indicated indefinite integrals. a. ∫ (x2 + x) dx b. ∫ (x3 + √x) dx c. ∫ (x + 1)2 dx
In Problems a-b, use the methods of Examples 5 and 6 to evaluate the indefinite integrals. a. ∫ (√2x + 1)3 √2 dx b. ∫ (πx3 + 1)4 3πx2 dx
In Problems a-b, f"(x) is given. Find f (x) by antidifferentiating twice. Note that in this case your answer should involve two arbitrary constants, one from each antidifferentiation. For example, if
Prove the formula ∫[f(x) g'(x) + g(x) f'(x)] dx = f(x) g(x) + C
Prove the formula
Use the formula from Problem 43 to find
Use the formula Problem 43 to find
Find ∫ f''(x) dx if f(x) = x √x3 + 1.
Prove the formula

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