The president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its first and second derivatives.
> A fiberglass pipe is lifted by a sling, as shown in the figure. The outer diameter of the pipe is 6.0 in., its thickness is 0.25 in, and its weight density is 0.053 lb/in3. The length of the pipe is L = 36 ft and the distance between lifting points is s
> A railroad tie (or sleeper) is subjected to two rail loads, each of magnitude P = 175 kN, acting as shown in the figure. The reaction q of the ballast is assumed to be uniformly distributed over the length of the tie, which has cross-sectional dimensions
> A curved bar ABC having a circular axis (radius r = 12 in.) is loaded by forces P = 400 lb (see figure). The cross section of the bar is rectangular with height h and thickness t. If the allowable tensile stress in the bar is 12,000 psi and the height h
> The horizontal beam ABC of an oil-well pump has the cross section shown in the figure. If the vertical pumping force acting at end C is 9 kips and if the distance from the line of action of that force to point B is 16 ft, what is the maximum bending stre
> During construction of a highway bridge, the main girders are cantilevered outward from one pier toward the next (see figure). Each girder has a cantilever length of 48 m and an I-shaped cross section with dimensions shown in the figure. The load on each
> A seesaw weighing 3 lb/ft of length is occupied by two children, each weighing 90 lb (see figure). The center of gravity of each child is 8 ft from the fulcrum. The board is 19 ft long, 8 in. wide, and 1.5 in. thick. What is the maximum bending stress in
> A freight-car axle AB is loaded approximately as shown in the figure, with the forces P representing the car loads (transmitted to the axle through the axle boxes) and the forces R representing the rail loads (transmitted to the axle through the wheels).
> Each girder of the lift bridge (see figure) is 180 ft long and simply supported at the ends. The design load for each girder is a uniform load of intensity 1.6 kips/ft. The girders are fabricated by welding three steel plates to form an I-shaped cross se
> Beam ABC has simple supports at A and B and an overhang from B to C. The beam is constructed from a steel W 16 × 31. The beam must carry its own weight in addition to uniform load q = 150 lb/ft. Determine the maximum tensile and compressive
> A simply supported wood beam AB with a span length L = 4 m carries a uniform load of intensity q = 5.8 kN/m (see figure). (a) Calculate the maximum bending stress σmax due to the load q if the beam has a rectangular cross section with width b
> A thin, high-strength steel rule (E = 30 × 106 psi) having a thickness t = 0.175 in and length L = 48 in is bent by couples Mo into a circular arc subtending a central angle α = 40° (see figure). (a) What is the maxi
> A steel wire (E = 200 GPa) of a diameter d = 1.25 mm is bent around a pulley of a radius Ro = 500 mm (see figure). (a) What is the maximum stress σmax in the wire? (b) By what percent does the stress increase or decrease if the radius of the p
> An aluminum pole for a street light weighs 4600Â N and supports an arm that weighs 660 N (see figure). The center of gravity of the arm is 1.2 m from the axis of the pole. A wind force of 300 N also acts in the (2y) direction at 9 m above the
> The graph of the derivative f 9 of a function f is shown. (a) On what intervals is f increasing or decreasing? (b) At what values of x does f have a local maximum or minimum? y y= f'(x) 2 4 6
> 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 curve in one family is orthogonal to every curve in the oth
> The graph of the derivative f 9 of a function f is shown. (a) On what intervals is f increasing or decreasing? (b) At what values of x does f have a local maximum or minimum? y y = f'(x) + 2 4 6.
> Sketch the graph of a function that satisfies all of the given conditions. (a) f ‘(x) > 0 and f ‘‘(x) < 0 for all x (b) f ‘(x) < 0 and f ‘‘(x) > 0 for all x
> (a) State the First Derivative Test. (b) State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?
> Suppose you are given a formula for a function f. (a) How do you determine where f is increasing or decreasing? (b) How do you determine where the graph of f is concave upward or concave downward? (c) How do you locate inflection points?
> Use the given graph of f to find the following. (a) The open intervals on which f is increasing. (b) The open intervals on which f is decreasing. (c) The open intervals on which f is concave upward. (d) The open intervals on which f is concave downward.
> Use the given graph of f to find the following. (a) The open intervals on which f is increasing. (b) The open intervals on which f is decreasing. (c) The open intervals on which f is concave upward. (d) The open intervals on which f is concave downward.
> The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions f, g, and h whose values at 0 are all 0 and, for x ≠0, (a) Show that 0 is a critica
> Suppose f is differentiable on an interval I and f ‘(x) > 0 for all numbers x in I except for a single number c. Prove that f is increasing on the entire interval I.
> Suppose that f ’’’ is continuous and f ‘(c) = f ’’(c) = 0, but f ’’’(c) > 0. Does f have a local maximum or minimum at c? Does f have a point of inflection at c?
> Show that the function g(x) = x |x | has an inflection point at (0, 0) but g ’’(0) does not exist.
> (a) Use the Product Rule twice to prove that if f , g, and h are differentiable, then s (fgh)’ = f’ gh +fg’ h + fgh’ . (b) Taking f = g = h in part (a), show that (c) Use part (b) to
> Sketch the graph of a function that satisfies all of the given conditions. (a) f ‘(x) < 0 and f ‘‘(x) < 0 for all x (b) f ‘(x) > 0 and f ‘‘(x) > 0 for all x
> Show that if f (x) = x4, then f ’’(0) = 0, but (0, 0) is not an inflection point of the graph of f .
> Prove that if (c, f (c)) is a point of inflection of the graph of f and f ’’ exists in an open interval that contains c, then f ’’(c) = 0.
> For what values of c does the polynomial P(x) = x4 + cx3 + x2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases?
> Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three x-intercepts x1, x2, and x3, show that the x-coordinate of the inflection point is (x1 + x2 + x3)/3.
> Show that tan x > x for 0 < x < π/2.
> Suppose f and g are both concave upward on (-∞,∞). Under what condition on f will the composite function h(x) = f (g(x)) be concave upward?
> (a) If f and g are positive, increasing, concave upward functions on I, show that the product function fg is concave upward on I. (b) Show that part (a) remains true if f and g are both decreasing. (c) Suppose f is increasing and t is decreasing. Show, b
> Assume that all of the functions are twice differentiable and the second derivatives are never 0. (a) If f and g are concave upward on I, show that f + g is concave upward on I. (b) If f is positive and concave upward on I, show that the function g(x) =
> Show that the inflection points of the curve y = x sin x lie on the curve y2(x2 + 4) = 4x2.
> Suppose f ‘ is continuous on (-∞, ∞). (a) If f (2) = 0 and f ’’(2) = -5, what can you say about f ? (b) If f ‘(6) = 0 and f ’’(6) = 0, what can you say about f ?
> 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 curve in one family is orthogonal to every curve in the oth
> Show that the curves y = e-x and y = -e-x touch the curve y = e-x sin x at its inflection points.
> Show that the curve y = (1 + x)/(1 + x2) has three points of inflection and they all lie on one straight line.
> For what values of a and b is (2, 2.5) an inflection point of the curve x2y + ax + by = 0? What additional inflection points does the curve have?
> (a) If the function f (x) = x3 + ax2 + bx has the local minimum value − 2 9 3 at x = 1/ 3 , what are the values of a and b? (b) Which of the tangent lines to the curve in part (a) has the smallest slope?
> Find a cubic function f (x) = ax3 + bx2 + cx + d that has a local maximum value of 3 at x = -2 and a local minimum value of 0 at x = 1.
> A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t) = Atpe-kt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decl
> (a) Find the critical numbers of f (x) = x4(x – 1)3. (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? (c) What does the First Derivative Test tell you?
> Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time. Account for the shape of the graph in terms of concavity. Wha
> Let K(t) be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, K(8) – K(7) or K(3) – K(2)? Is the graph of K concave upward or concave downward? Why?
> 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 curve in one family is orthogonal to every curve in the oth
> Let f (t) be the temperature at time t where you live and suppose that at time t = 3 you feel uncomfortably hot. How do you feel about the given data in each case? (a) f ‘(3) = 2, f ‘(3) = 4 (b) f ‘(3) = 2, f ‘(3) = 24 (c) f ‘(3) = 22, f ‘(3) = 4 (d) f ‘
> In an episode of The Simpsons television show, Homer reads from a newspaper and announces “Here’s good news! According to this eye-catching article, SAT scores are declining at a slower rate.” Interpret Homer’s statement in terms of a function and its fi
> A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward o
> Use the methods of this section to sketch the curve y = x3 - 3a2x + 2a3, where a is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?
> Suppose the derivative of a function f is f (x) = (x + 1)2 (x – 3)5 (x – 6)4. On what interval is f increasing?
> (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. f(x) = x2 - x - ln x
> Prove the formula for (d/dx)(cos-1x) by the same method as for (d/dx)(sin-1x).
> (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. f(x) = x2 ln x
> (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. f(x) = e2x + e-x
> (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. f(x) =cos2x - 2 sin x, 0 ≤ x ≤ 2π
> (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. f(x) =sin x + cos x, 0 ≤ x ≤ 2 π
> (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. f(x) = x4 - 2x2 + 3
> (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. f(x) = 2x3 - 9x2 + 12x - 3
> (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points. f(x) = x3 - 3x2 - 9x + 4
> The graph of the first derivative f ‘ of a function f is shown. (a) On what intervals is f increasing? Explain. (b) At what values of x does f have a local maximum or minimum? Explain. (c) On what intervals is f concave upward or concav
> In each part state the x-coordinates of the inflection points of f. Give reasons for your answers. (a) The curve is the graph of f. (b) The curve is the graph of f ‘. (c) The curve is the graph of f ’’
> Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. f(x) = sin (x/2), [π/2, 3π/2]
> Draw the graph of a function that is continuous on [0, 8] where f (0) = 1 and f (8) = 4 and that does not satisfy the conclusion of the Mean Value Theorem on [0, 8].
> The graph of a function t is shown. (a) Verify that t satisfies the hypotheses of the Mean Value Theorem on the interval [0, 8]. (b) Estimate the value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval [0, 8]. (c) Estimate t
> Draw the graph of a function defined on [0, 8] such that f (0) = f (8) = 3 and the function does not satisfy the conclusion of Rolle’s Theorem on [0, 8].
> The graph of a function f is shown. Verify that f satisfies the hypotheses of Rolle’s Theorem on the interval [0, 8]. Then estimate the value(s) of c that satisfy the conclusion of Rolle’s Theorem on that interval.
> A number a is called a fixed point of a function f if f (a) = a. Prove that if f ‘(x) ≠ 1 for all real numbers x, then f has at most one fixed point.
> Find the derivative of the function. Simplify where possible. y = cos-1(sin-1 t)
> Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed.
> At 2:00 pm a car’s speedometer reads 30 mi/h. At 2:10 pm it reads 50 mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h2.
> Use the method of Example 6 to prove the identity 2 sin-1x = cos-1(1 - 2x2) x ≥ 0 Example 6: The function f (x) = |x | has its (local and absolute) minimum value at 0, but that value can’t be found by setting f â
> Let f (x) = 1/x and Show that f 9sxd − t9sxd for all x in their domains. Can we conclude from Corollary 7 that f - g is constant? Corollary 7: 1 if x>0 g(x) = 1 if x<0 1 + 7 Corollary If f'(x) = g'(x) for all x in an interval (a, b
> If f ‘(x) = c (c a constant) for all x, use Corollary 7 to show that f (x) = cx + d for some constant d. Corollary 7: 7 Corollary If f'(x) = g'(x) for all x in an interval (a, b), then f – g is constant on (a, b); that is, f(x) = g
> Use the Mean Value Theorem to prove the inequality |sin a - sin b | ≤ |a - b | for all a and b
> Suppose f is an odd function and is differentiable everywhere. Prove that for every positive number b, there exists a number c in s(-b, b) such that f ‘(c) = f (b)/b.
> Show that sin x < x if 0 < x < 2π .
> Suppose that f and t are continuous on [a, bg] and differentiable on [a, b]. Suppose also that f (a) = g (a) and f’ (x) = g’ (x) for a < x < b. Prove that f (b) < g (b).
> Find the derivative of the function. Simplify where possible. y = x sin-1 x + 1− x2
> Does there exist a function f such that f (0) = -1, f (2) = 4, and f’ (x) ≤ 2 for all x?
> Suppose that 3 ≤ f ‘(x) ≤ 5 for all values of x. Show that 18 ≤ f (8) - f (2) ≤ 30.
> If f (1) = 10 and f ‘(x) ≥ 2 for 1 ≤ x ≤ 4, how small can f (4) possibly be?
> (a) Suppose that f is differentiable on R and has two roots. Show that f ‘ has at least one root. (b) Suppose f is twice differentiable on R and has three roots. Show that f ’’ has at least one real root. (c) Can you generalize parts (a) and (b)?
> (a) Show that a polynomial of degree 3 has at most three real roots. (b) Show that a polynomial of degree n has at most n real roots.
> Show that the equation x4 + 4x + c = 0 has at most two real roots.
> Show that the equation x3 - 15x + c = 0 has at most one root in the interval [-2, 2].
> Show that the equation has exactly one real root. x3 + ex = 0
> Show that the equation has exactly one real root. 2x + cos x = 0
> Let f (x) = 2 - |2x - 1|. Show that there is no value of c such that f (3) - f (0) = f ‘(c)(3 – 0). Why does this not contradict the Mean Value Theorem?
> The biomass B(t) of a fish population is the total mass of the members of the population at time t. It is the product of the number of individuals N(t) in the population and the average mass M(t) of a fish at time t. In the case of guppies, breeding occu
> Let f (x) = ( x – 3)-2. Show that there is no value of c in (1, 4) such that f (4) - f (1) = f ‘(c)(4 – 1). Why does this not contradict the Mean Value Theorem?
> Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at (c, f(c)). Are the secant line and the tangent line parallel? f(x) = e-x ,
> Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. f (x) = 1/x, [1, 3]
> Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. f (x) = ln x, [1, 4]
> Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. f (x) = x3 - 3x + 2, [-2, 2]
> Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. f (x) = 2x2 - 3x + 1, [0, 2]
> Let f (x) = tan x. Show that f (0) = f ( π) but there is no number c in (0, π) such that f ‘(c) = 0. Why does this not contradict Rolle’s Theorem?
> Let f (x) = 1 - x2/3. Show that f (-1) = f (1) but there is no number c in (-1, 1) such that f ‘(c) = 0. Why does this not contradict Rolle’s Theorem?
