Show that if f (x) = x4, then f ’’(0) = 0, but (0, 0) is not an inflection point of the graph of f .
> A cantilever beam AB is loaded by a couple 0M at its free end (see figure). The length of the beam is L = 2.0 m, and the longitudinal normal strain at the top surface is ε = 0.0010. The distance from the top surface of the beam to the neutra
> A 4.75-in. outside diameter polyethylene pipe designed to carry chemical waste is placed in a trench and bent around a quarter-circular 908 bend (see figure). The bent section of the pipe is 52 ft long. (a) Determine the maximum compressive strain Î
> A copper wire having a diameter of d = 4mm is bent into a circle and held with the ends just touching (see figure). (a) If the maximum permissible strain in the copper is εmax = 0.0024, what is the shortest length L of wire that can be used?
> A palm tree weighing 1000 lb is inclined at an angle of 60° (see figure). The weight of the tree may be resolved into two resultant forces: a force P1 = 900 lb acting at a point 12 ft from the base and a force P2 = 100 lb acting at the top of
> A steel wire with a diameter of d = 1/16 in is bent around a cylindrical drum with a radius of R = 36 in (see figure). (a) Determine the maximum normal strain εmax. (b) What is the minimum acceptable radius of the drum if the maximum normal
> Beam ABCDE has a moment release just right of joint B and has concentrated moment loads at D and E. In addition, a cable with tension P is attached at F and runs over a pulley at C (Fig. a). The beam is constructed using two steel plates, which are welde
> A steel post (E = 3 0 × 106 psi) having thickness t = 1/8 in and height L = 72 in supports a stop sign (see figure), where s = 12.5 in. The height of the post L is measured from the base to the centroid of the sign. The stop sign is subjecte
> A foot bridge on a hiking trail is constructed using two timber logs each having a diameter d = 0.5 m (see figure a). The bridge is simply supported and has a length L = 4 m. The top of each log is trimmed to form the walking surface (see Fig. b). A simp
> A small dam of a height h = 6 ft is constructed of vertical wood beams AB, as shown in the figure. The wood beams, which have a thickness t = 2.5 in, are simply supported by horizontal steel beams at A and B. Construct a graph showing the maximum bending
> Consider the compound beam with segments AB and BCD joined by a pin connection (moment release) just right of B (see figure part a). The beam cross section is a double-T made up from three 50mm × 150 mm wood members (actual dimensions, see f
> A beam with a T-section is supported and loaded as shown in the figure. The cross section has width b = 21/2 in, height h = 3 in, and thickness t = 3/8 in. (a) Determine the maximum tensile and compressive stresses in the beam. (b) If the allowable stre
> A cantilever beam AB with a rectangular cross section has a longitudinal hole drilled throughout its length (see figure). The beam supports a load P = 600 N. The cross section is 25 mm wide and 50 mm high, and the hole has a diameter of 10 mm. Find the b
> A beam ABC with an overhang from B to C supports a uniform load of 200 lb/ft throughout its length (see figure). The beam is a channel section with dimensions as shown in the figure. The moment of inertia about the z axis (the neutral axis) equals 8.13 i
> A rigid frame ABC is formed by welding two steel pipes at B (see figure). Each pipe has cross- sectional area A = 11.31 × 103 mm2, moment of inertia I = 46.37 × 106 mm4, and outside diameter d = 200 mm Find the maximum tensile a
> A cantilever beam, a C12 × 30 section, is subjected to its own weight and a point load at B. Find the maximum permissible value of load P at B (kips) if the allowable stress in tension and compression is σa = 18 ksi. P q = 30
> A cantilever beam AB of an isosceles trapezoidal cross section has a length L = 0.8 m, dimensions b1 = 80 mm and b2 = 90 mm, and height h = 110 mm (see figure). The beam is made of brass weighing 85 kN/m3. (a) Determine the maximum tensile stress Ï&
> A cantilever beam AB, loaded by a uniform load and a concentrated load (see figure), is constructed of a channel section. (a) Find the maximum tensile stress σt and maximum compressive stress σc if the cross section has the dimensio
> Determine the maximum tensile stress σt and maximum compressive stress σc due to the load P acting on the simple beam AB (see figure). (a) Data are P = 6.2 kN, L = 3.2m, d = 1.25 m, b = 80 mm, t = 25 mm, h = 120 mm, and h1 = 90 mm.
> A simple beam AB of a span length L = 24 ft is subjected to two wheel loads acting at a distance d = 5 ft apart (see figure). Each wheel transmits a load P = 3.0 kips, and the carriage may occupy any position on the beam. (a) Determine the maximum bendin
> Determine the maximum bending stress σmax (due to pure bending by a moment M) for a beam having a cross section in the form of a circular core (see figure). The circle has diameter d and the angle β = 60°. -d-
> Determine the maximum tensile stress σt (due to pure bending about a horizontal axis through C by positive bending moments M) for beams having cross sections as follows (see figure). (a) A semicircle of diameter d. (b) An isosceles trapezoid w
> A small dam of height h = 2.0 m is constructed of vertical wood beams AB of thickness t = 120 mm, as shown in the figure. Consider the beams to be simply supported at the top and bottom. Determine the maximum bending stress σmax in the beams,
> 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
> 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 ‘
> 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.
> 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.
