Find the derivative of the function. Simplify where possible. y = cos-1(sin-1 t)
> 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 ‘
> 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.
> 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?
> Find the derivative of the function. Simplify where possible. R(t) = arcsin(1/t)
> For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum. b c d d r s x a
> For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum. yA 0 a b c d r
> Suppose f is a continuous function defined on a closed interval [a, b]. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f ? (b) What steps would you take to find those maximum and minimum values?
> Explain the difference between an absolute minimum and a local minimum.
> A cubic function is a polynomial of degree 3; that is, it has the form f (x) = ax3 + bx2 + cx + d, where a ≠ 0. (a) Show that a cubic function can have two, one, or no critical number(s). Give examples and sketches to illustrate the three possibilities.
> Prove Fermat’s Theorem for the case in which f has a local minimum at c.
> If f is the function considered in Example 3, use a computer algebra system to calculate f ‘ and then graph it to confirm that all the maximum and minimum values are as given in the example. Calculate f ’â€&
> (a) Graph the function. (b) Use l’Hospital’s Rule to explain the behavior as x ( 0. (c) Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values. f(x) = x e1/x
> (a) Graph the function. (b) Use l’Hospital’s Rule to explain the behavior as x ( 0. (c) Estimate the minimum value and intervals of concavity. Then use calculus to find the exact values. f(x) = x2 ln x
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = ex - 0.
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = 6 sin x
> Use the asymptotic behavior of f (x) = sin x + e-x to sketch its graph without going through the curve sketching procedure of this section.
> Show that the lines y = (b/a)x and y = -(b/a)x are slant asymptotes of the hyperbola (x2/a2) – (y2/b2) = 1.
> (a) Find y’ by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y’ in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part
> Show that the curve y = x – tan-1x has two slant asymptotes: y = x + π/2 and y = x - π/2. Use this fact to help sketch the curve.
> (a) Find y’ by implicit differentiation. (b) Solve the equation explicitly for y and differentiate to get y’ in terms of x. (c) Check that your solutions to parts (a) and (b) are consistent by substituting the expression for y into your solution for part
> A model for the concentration at time t of a drug injected into the bloodstream is C(t) = K(e-at – e-bt) where a, b, and K are positive constants and b > a. Sketch the graph of the concentration function. What does the graph tell us about how the concent
> The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region x2 + 4y2 ≤ 5. If the point (-5, 0) is on the edge of the shadow, how far above the x-axis is the lamp located?
> Find f ‘(x). Check that your answer is reasonable by comparing the graphs of f and f ‘. f(x) = arctan (x2 – x)
> The Bessel function of order 0, y = J(x), satisfies the differential equation xy’’ + y’ + xy = 0 for all values of x and its value at 0 is J(0) = 1. (a) Find J’(0). (b) Use implicit differentiation to find J’’(0).
> Use the guidelines of this section to sketch the curve. y = (1 – x)ex
