Differentiate the function. y = [ln (1 + ex)]2
> 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 -axis and show that the tangent lines at these points are parallel.
> Suppose that three points on the parabola y = x2 have the property that their normal lines intersect at a common point. Show that the sum of their x-coordinates is 0.
> Find the two points on the curve y = x4 – 2x2 – x that have a common tangent line.
> Given an ellipse x2/a2 + y2/b2 = 1, where a ≠ b, find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals.
> Suppose that we replace the parabolic mirror of Problem 18 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle with center O. A ray of light coming in toward the mir
> A car is traveling at night along a highway shaped like a parabola with its vertex at the origin. The car starts at a point 100 m west and 100 m north of the origin and travels in an easterly direction. There is a statue located 100 m east and 50 m north
> (a). Use the identity for tan (x – y) (see Equation 14b in Appendix C) to show that if two lines L1 and L2 intersect at an angle a, then tan a = m2 – m1/1 + m1m2 where m1 and m2 are the slopes of L1 and L2, respectivel
> (a). The cubic function f (x) = x (x – 2) (x – 6) has three distinct zeros: 0, 2, and 6. Graph f and its tangent lines at the average of each pair of zeros. What do you notice? (b). Suppose the cubic function f (x) = x (x – a) (x – b) (x – c) has three d
> If a stone is thrown vertically upward from the surface of the moon with a velocity of 10 m/s, its height (in meters) after seconds is h = 10t – 0.83t2. (a). What is the velocity of the stone after 3 s? (b). What is the velocity of the stone after it has
> If a ball is given a push so that it has an initial velocity of 5 m/s down a certain inclined plane, then the distance it has rolled after seconds is s = 5t + 3t2. (a). Find the velocity after 2 s. (b). How long does it take for the velocity to reach 35
> The position function of a particle is given by s = t3 – 4.5t2 – 7t, t > 0. (a). When does the particle reach a velocity of 5 m/s? (b). When is the acceleration 0? What is the significance of this value of t?
> Let f (x) = x/√1 – cos 2x (a). Graph f. What type of discontinuity does it appear to have at 0? (b). Calculate the left and right limits of f at 0. Do these values confirm your answer to part (a)?
> Graphs of the velocity functions of two particles are shown, where is measured in seconds. When is each particle speeding up? When is it slowing down? Explain. (a) (b) ৮
> Differentiate the function. f (x) = log5 (xex)
> Differentiate the function. f (x) = log2 (1 – 3x)
> Differentiate the function. f (x) = ln (sin2 x)
> The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV = nRT, where n is the number of moles of the gas and R = 0.0821 is the gas constant. Suppose that, at a certain instant, P =
> Let f (x) = loga (3x2 – 2). For what value of a is f'(1) = 3?
> If p (x) is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is A (x) = p (x)/x. (a). Find A'(x). Why does the company want to hire more workers if A'(x) > 0? (b). Show tha
> The cost function for production of a commodity is C (x) = 339 + 25x – 0.09x2 + 0.0004x3 (a). Find and interpret C"(100). (b). Compare C"(100) with the cost of producing the 101st item.
> (a). On what interval is f (x) = x ln x decreasing? (b). On what interval is f concave upward?
> Find equations of the tangent lines to the curve y = (lnx)/x at the points (1, 0) and (e, 1/e). Illustrate by graphing the curve and its tangent lines.
> The table shows how the average age of first marriage of Japanese women varied in the last half of the 20th century. (a). Use a graphing calculator or computer to model these data with a fourth-degree polynomial. (b). Use part (a) to find a model for A
> Show, using implicit differentiation, that any tangent line at a point P to a circle with center O is perpendicular to the radius OP.
> Find an equation of the tangent line to the curve at the given point. y = ln (x2 - 3x + 1), (3, 0)
> The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function where is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 12 cell
> Differentiate f and find the domain of f. f (x) = x/1 – ln (x – 1)
> Find y' and y". y = x2 ln (2x)
> Differentiate the function. y = log2 (e-x cos πx)
> Differentiate the function. f (x) = x ln x - x
> Differentiate the function. y = 2x log10√x
> The mass of the part of a metal rod that lies between its left end and a point meters to the right is 3x2 kg. Find the linear density (see Example 2) when is (a) 1 m, (b) 2 m, and (c) 3 m. Where is the density the highest? The lowest?
> A spherical balloon is being inflated. Find the rate of increase of the surface area (S = π4r2) with respect to the radius when is (a) 1 ft, (b) 2 ft, and (c) 3 ft. What conclusion can you make?
> A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after (a) 1 s, (b) 3 s, and (c) 5 s. What can you conclude?
> (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)?
> Differentiate the function. g (x) = ln (x √x2 – 1)
> Differentiate the function. h (x) = ln (x + √x2 – 1)
> Differentiate the function. F (t) = ln (2t + 1)3/ (3t – 1)4
> Differentiate the function. f (t) = x 1 + ln t/1 – ln t
> Differentiate the function. f (x) = sin x ln (5x)
> Differentiate the function. f (x) = ln 5√x
> Differentiate the function. f (x) = 5√ln x
> Use logarithmic differentiation to find the derivative of the function. y = (2x + 1)5(x4 – 3)6
> Let f (x) = cx + ln (cos x). For what value of c is f'(π/4) = 6?
> If f (x) = sin x + ln x, find f'(x). Check that your answer is reasonable by comparing the graphs of f and f'.
> Estimate the value of f'(a) by zooming in on the graph off. Then differentiate f to find the exact value of f'(a) and compare with your estimate. f (x) = 3x2 - x3, a = 1
> Find an equation of the tangent line to the curve at the given point. y = ln (x3 – 7), (2, 0)
> Differentiate f and find the domain of f. f (x) = ln ln ln x
> Find y' and y". y = ln x/x2
> Differentiate the function. y = ln (e-x + xe-x)
> Differentiate the function. y = ln |2 – x = 5x2|
> Differentiate the function. F (y) = y ln (1 + ey)
> Prove that cos (sin-1 x) = √1 – x2.
> Find the exact value of each expression. cos (tan-1 2 + tan-13)
> Find the exact value of each expression. sin (2 tan-1 √2)
> (a). Evaluate limx→∞ x sin 1/x. (b). Evaluate limx→0 x sin 1/x. (c). Illustrate parts (a) and (b) by graphing y = x sin (1/x).
> Find the exact value of each expression. csc (arccsc3/5)
> Find the exact value of each expression. tan (sin-1 (2/3)
> A semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a two-dimensional ice-cream cone, as shown in the figure. If A (θ) is the area of the semicircle and B (θ)is the area of the triangle,
> (a). The van der Waals equation for n moles of a gas is where P is the pressure V, is the volume, and T is the temperature of the gas. The constant R is the universal gas constant and a and b are positive constants that are characteristic of a particular
> Find the value of the number a such that the families of curves y = (x + c)-1 and y = a (x + k)1/3 are orthogonal trajectories.
> 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).
> (a). Sketch the graph of the function f (x) = sin (sin-1 x). (b). Sketch the graph of the function g (x) = sin-1 (sin x), x ∈ R (c). Show that g'(x) = cos x/|cos x|. (d). Sketch the graph of h (x) = cos-1 (sin x), x ∈ R, and find its derivative.
> Use the formula from Exercise 41(a) to prove Exercise 41(a): (a). Suppose f is a one-to-one differentiable function and its inverse function f-1 is also differentiable. Use implicit differentiation to show that (f-1)'(x) = 1/f'(f-1(x)) provided that th
> (a). Show that f (x) = 2x + cos x is one-to-one. (b). What is the value of f-1(1)? (c). Use the formula from Exercise 41(a) to find (f-1)'(1). Exercise 41(a): (a). Suppose f is a one-to-one differentiable function and its inverse function f-1 is also d
> (a). Suppose f is a one-to-one differentiable function and its inverse function f-1 is also differentiable. Use implicit differentiation to show that (f-1)'(x) = 1/f'(f-1(x)) provided that the denominator is not 0. (b). If f (4) = 5 and f'(4) = 2/3, find
> If f and g are the functions whose graphs are shown, let u (x) = f (x) g (x) and v (x) = f (x) /g (x). (a). Find u'(1) (b). Find v'(5)
> Find the limit. limx→0+ tan-1 (ln x)
> Find the exact value of each expression. (a). tan-1 (tan3π/4) (b). cos (arccos ½)
> Find the limit. limx→∞+ arctan (ex)
> Find the limit. limx→∞ arccos (1 + x2/1 + 2x2)
> Find the limit. Limx→-1+ sin-1 x
> Find f'(x). Check that your answer is reasonable by comparing the graphs of f and f'. f (x) = arctan (x2 -x)
> Find f'(x). Check that your answer is reasonable by comparing the graphs of f and f'. f (x) = √1 - x2 arcsin x
> Find an equation of the tangent line to the curve y = 3 arccos (x/2) at the point (1, π).
> If g (x) = x sin (x/4) + √16 – x2, find g'(2).
> Find y' if tan-1 (xy) = 1 + x2y.
> The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices. у. a b
> Find the derivative of the function. Find the domains of the function and its derivative. g (x) = c0s-1 (3 – 2x)
> Find the derivative of the function. Find the domains of the function and its derivative. f (x) = arcsin (ex)
> Find the exact value of each expression. (a). arctan 1 (b). sin-1 (1/√2)
> Find the derivative of the function. Simplify where possible. y = arccos (b + a cos x/a + b cos x), 0 < x < π, a > b > 0
> Find the derivative of the function. Simplify where possible. y = arctan √1-x/1+x
> Find the derivative of the function. Simplify where possible. y = x sin-1 x + √1 – x2
> Find the derivative of the function. Simplify where possible. y = cos-1 (sin-1 t)
> Find the derivative of the function. Simplify where possible. y = arctan (cos θ)
> Find the derivative of the function. Simplify where possible. y = tan-1 (x – √1 + x2)
> Find the derivative of the function. Simplify where possible. y = cos-1(e2x)
> The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices. y b
> Find the derivative of the function. Simplify where possible. f (x) = x ln (arctan x)
> Find the derivative of the function. Simplify where possible. G (x) = √1 – x2 arccos x
> Find the derivative of the function. Simplify where possible. F (θ) = arcsin √sin θ
> Find the exact value of each expression. (a). tan-1 (1/√3) (b). sec-1 (2)
> Find the derivative of the function. Simplify where possible. y = sin-1 (2x + 1)
> Find the derivative of the function. Simplify where possible. y = tan-1 (x2)
> Find the derivative of the function. Simplify where possible. y = (tan-1 x)2
> (a). Prove that sin-1 x = π/2. (b). Use part (a) to prove Formula 2. sin1x cos1x
> Prove Formula 2 by the same method as for Formula 1.
> Graph the given functions on the same screen. How are these graphs related? y = tan x, - 7/2 < x < m/2; y= tan='x; y = x
> The figure shows graphs of f, f', f'' and f'''. Identify each curve, and explain your choices. a b c d
