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?
> 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?
> 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
> 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
> Use the guidelines of this section to sketch the curve. y = arctan(ex)
> Use the guidelines of this section to sketch the curve. y = csc - 2sin x, 0 < x < π
> Use the guidelines of this section to sketch the curve. y = 2x - tan x, -π/2 < x < π/2
> Use the guidelines of this section to sketch the curve. y = x tan x, -π/2 < x < π/2
> (a) Show that f (x) = x + ex is one-to-one. (b) What is the value of f-1 (1)? (c) Use the formula from Exercise 77(a) to find s (f-1)’(1). Data from Exercise 77(a): (a) Suppose f is a one-to-one differentiable function and its inverse function f-1 is al
> The graph of the derivative f ‘ of a continuous function f is shown. (a) On what intervals is f increasing? Decreasing? (b) At what values of x does f have a local maximum? Local minimum? (c) On what intervals is f concave upward? Conca
> (a) If F(x) = f (x) g(x), where f and g have derivatives of all orders, show that F’’ = f ‘’g + 2f’ g’ + f g’’. (b) Find similar formulas for F’’’ and F(4). (c) Guess a formula for F(n).
> The graph of the derivative f ‘ of a continuous function f is shown. (a) On what intervals is f increasing? Decreasing? (b) At what values of x does f have a local maximum? Local minimum? (c) On what intervals is f concave upward? Conca
> Find equations of both the tangent lines to the ellipse x2 + 4y2 = 36 that pass through the point (12, 3).
> The graph of a function y = f (x) is shown. At which point(s) are the following true? (a) dy/dx and d2y/dx2 are both positive. (b) dy/dx and d2y/dx2 are both negative. (c) dy/dx is negative but d2y/dx2 is positive. y. D E А B
> Let (a) Show that f is continuous at 0. (b) Investigate graphically whether f is differentiable at 0 by zooming in several times toward the point (0, 1) on the graph of f . (c) Show that f is not differentiable at 0. How can you reconcile this fact with
> Let (a) Use the definition of derivative to compute f ‘(0). (b) Show that f has derivatives of all orders that are defined on R. le-/ if x 0 f(x) = if x = 0
> Find all points on the curve x2y2 + xy = 2 where the slope of the tangent line is -1.
> Suppose f is a continuous function where f (x) > 0 for all x, f (0) = 4, f ‘(x) > 0 if x < 0 or x > 2, f ‘(x) < 0 if 0 < x < 2, f ’’(-1) = f ’’(1) = 0, f ’’(x) > 0 if x < -1 or x > 1, f ’’(x) < 0 if -1 < x < 1. (a) Can f have an absolute maximum? If so,
> Investigate the family of curves f (x) = ex - cx. In particular, find the limits as x ( ±∞ and determine the values of c for which f has an absolute minimum. What happens to the minimum points as c increases?
> Suppose f (3) = 2, f ‘(3) = 1/2 , and f ’(x) > 0 and f ’’(x) < 0 for all x. (a) Sketch a possible graph for f. (b) How many solutions does the equation f (x) = 0 have? Why? (c) Is it possible that f ‘(2) = 1/3 ? Why?
> (a) Where does the normal line to the ellipse x2 - xy + y2 = 3 at the point (-1, 1) intersect the ellipse a second time? (b) Illustrate part (a) by graphing the ellipse and the normal line.
> Illustrate l’Hospital’s Rule by graphing both f (x)/g(x) and f ’(x)/g’ (x) near x = 0 to see that these ratios have the same limit as x ( 0. Also, calculate the exact value of the limit. f (x) = 2x sin x, g(x) = sec x - 1
> Illustrate l’Hospital’s Rule by graphing both f (x)/g(x) and f ’(x)/g’ (x) near x = 0 to see that these ratios have the same limit as x ( 0. Also, calculate the exact value of the limit. f (x) = ex - 1, g(x) = x3 + 4x
> The equation x2 - xy + y2 = 3 represents a “rotated ellipse,” that is, an ellipse whose axes are not parallel to the coordinate axes. Find the points at which this ellipse crosses the x-axis and show that the tangent lines at these points are parallel.
> (a) Use implicit differentiation to find y’ if x2 + xy + y2 - 1 = 0 (b) Plot the curve in part (a). What do you see? Prove that what you see is correct. (c) In view of part (b), what can you say about the expression for y’ that you found in part (a)?
> Find the value of the number a such that the families of curves y = s(x + c)-1 and y = a(x + k)1/3 are orthogonal trajectories.
> Sketch the graph of a function that satisfies all of the given conditions. f ‘(x) > 0 for all x ≠ 1, vertical asymptote x = 1, f ’’(x) > 0 if x < 1 or x > 3, f ’’(x) < 0 if 1 < x < 3
> Show that the ellipse x2/a2 + y2/b2 = 1 and the hyperbola x2/A2 - y2/B2 = 1 are orthogonal trajectories if A2, a2 and a2 - b2 = A2 + B2 (so the ellipse and hyperbola have the same foci).
> Sketch the graph of a function that satisfies all of the given conditions. f ‘(0) = f ‘(2) = f ‘(4) = 0, f ‘(x) > 0 if x > 0 or 2 < x < 4, f ‘(x) < 0 if 0 < x < 2 or x > 4, f ‘‘(x) > 0 if 1 < x < 3, f ‘‘(x) < 0 if x < 1 or x > 3
> 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
> Sketch the graph of a function that satisfies all of the given conditions. Vertical asymptote x = 0, f ‘(x) > 0 if x < -2, f ‘(x) < 0 if x > -2 (x ≠ 0), f ‘‘(x) < 0 if x < 0, f ‘‘(x) > 0 if x > 0
> (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) = x4e-x
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = 2x3 - 3x2 - 12x + 1, [-2, 3]
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = 12 + 4x - x2, [0, 5]
> Find the critical numbers of the function. f(x) = x-2 ln x
> Find the derivative of the function. Simplify where possible. g(x) = arccos√x
> Find the critical numbers of the function. f(x) = x2 e-3x
