Find the critical numbers of the function. f(x) = 2x3 + x2 + 2x
> 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
> Find the critical numbers of the function. h(t) = 3t – arcsin t
> Find the critical numbers of the function. f(θ) = 2 cos θ + sin2 θ
> Find the critical numbers of the function. g(θ) = 4θ – tan θ
> A cup of hot chocolate has temperature 80°C in a room kept at 20°C. After half an hour the hot chocolate cools to 60°C. (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to 40°C?
> Find the critical numbers of the function. F(x) = x4/5 (x – 4)2
> Find the critical numbers of the function. h(t) = t3/4 – 2t1/4
> Find the critical numbers of the function. g(t) = |3t - 4 |
> Find the derivative of the function. Simplify where possible. y = sin-1(2x + 1)
> Find the critical numbers of the function. g(t) = t4 + t3 + t2 + 1
> Find the critical numbers of the function. f(x) = 2x3 – 3x2 – 36x
> Let C(t) be the concentration of a drug in the bloodstream. As the body eliminates the drug, C(t) decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus C’(t) = -kC(t), where k is a positive number called the
> Find the critical numbers of the function. f(x) = x3 + 6x2 – 15x
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) 2x + 1 if 0 <x< 1 f(x) 4 – 2x if 1 <I< 3
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) [x² f(x) = if -1 <x<0 if 0 <x<1 2 — Зх
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = ex
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = |x |
> Find the derivative of the function. Simplify where possible. y = tan-1(x2)
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = ln x, 0 , x ≤ 2
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (t) = cos t, -3 π/2 ≤ t ≤ 3 π/2
> Cobalt-60 has a half-life of 5.24 years. (a) Find the mass that remains from a 100-mg sample after 20 years. (b) How long would it take for the mass to decay to 1 mg?
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = sin x, - π/2 ≤ x ≤ π/2
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = sin x, 0 , x ≤ π/2
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = sin x, 0 ≤ x , π/2
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = 1/x, 1 < x < 3
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = 1/x, x ≥ 1
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = 2 – 1/3x, x ≥ -2
> Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.) f (x) = ½ (3x – 1), x ≤ 3
> (a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.
> Find the derivative of the function. Simplify where possible. y = (tan-1x)2
> (a) Sketch the graph of a function on [21, 2] that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on [21, 2] that is discontinuous but has both an absolute maximum and an absolute minimum.
> A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after t hours. (b) Find the number of bacteria after 4 hours. (c)
> (a) Sketch the graph of a function on [21, 2] that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on [21, 2] that has a local maximum but no absolute maximum.
> (a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2. (b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2. (c) Sketch the graph of a function that has a loc
> Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. Absolute maximum at 2, absolute minimum at 5, 4 is a critical number but there is no local maximum or minimum there.
> Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. Absolute minimum at 3, absolute maximum at 4, local maximum at 2
> Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. Absolute maximum at 4, absolute minimum at 5, local maximum at 2, local minimum at 3
> Sketch the graph of a function f that is continuous on [1, 5] and has the given properties. Absolute maximum at 5, absolute minimum at 2, local maximum at 3, local minima at 2 and 4
> Use the graph to state the absolute and local maximum and minimum values of the function. y= g(x) -1 1
> Use the graph to state the absolute and local maximum and minimum values of the function. yA y = f(x) 1 1
> Find an equation of the tangent line to the hyperbola x2 / a2 - y2 / b2 = 1 at the point (x0, y0).
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. An equation of the tangent line to the parabola y = x2 at (-2, 4) is y - 4 = 2x(x + 2).
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a rational function is a rational function.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. . If f (x) = (x6 – x4)5, then f (31)(x) = 0.
> The cost, in dollars, of producing x units of a certain commodity is C(x) = 920 + 2x - 0.02x2 + 0.00007x3 (a) Find the marginal cost function. (b) Find C’(100) and explain its meaning. (c) Compare C’(100) with the cost of producing the 101st item.
> Show, using implicit differentiation, that any tangent line at a point P to a circle with center O is perpendicular to the radius OP.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a polynomial is a polynomial.
> Show that the length of the portion of any tangent line to the astroid x2/3 + y2/3 = a2/3 cut off by the coordinate axes is constant.
> The Michaelis-Menten equation for the enzyme chymotrypsin is v = 0.14[S] / 0.015 + [S] where v is the rate of an enzymatic reaction and [S] is the concentration of a substrate S. Calculate dv/d[S] and interpret it.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If y = e2, then y’ = 2e.
> Suppose f is a differentiable function such that f (g(x)) = x and f ‘(x) = 1 + [f(x)]2 . Show that g’(x) = 1 / (1 + x2).
> Express the limit as a derivative and evaluate. lim ℎ→0 4 16+ℎ −2 ℎ
> A window has the shape of a square surmounted by a semi- circle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of th
> Evaluate dy if y = x3 - 2x2 + 1, x = 2, and dx = 0.2.
> The angle of elevation of the sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is π/6?
> Prove that the function f (x) = x101 + x51 + x + 1 has neither a local maximum nor a local minimum.
> A waterskier skis over the ramp shown in the figure at a speed of 30 ft/s. How fast is she rising as she leaves the ramp? 4 ft -15 ft–
> Show that 5 is a critical number of the function t(x) = 2 + (x – 5)3 but t does not have a local extreme value at 5.
> When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. This is accompanied by a contraction of the trachea, making a narrower channel for the expelled ai
> Find the derivative of the function. Simplify where possible. H(t) = cot-1(t) + cot-1(1/t)
> On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the velocity data for the shuttle between liftoff and the je
