Differentiate the following functions. y = ln (1/x)
> Evaluate the expression. Use ln 2 = .69 and ln 3 = 1.1. (a) ln 12 (b) ln 16 (c) ln(9 * 24)
> Evaluate the expression. Use ln 2 = .69 and ln 3 = 1.1. (a) ln 4 (b) ln 6 (c) ln 54
> Which is larger, ½ ln 16 or 13 ln 27? Explain.
> Which is larger, 2 ln 5 or 3 ln 3? Explain.
> Simplify the following expressions. ½ ln xy + 3/2 ln x/y
> Simplify the following expressions. ln x - ln x2 + ln x4
> Simplify the following expressions. e ln x2+3 ln y
> Simplify the following expressions. 5 ln x – ½ ln y + 3 ln z
> Simplify the following expressions. 3/2 ln 4 - 5 ln 2
> Simplify the following expressions. e2 ln x
> The following function may be viewed as a composite function h (x) = f ( g (x)). Find f (x) and g (x). h(x) = (4x - 3)3 + 1/4x - 3
> Figure 5 shows the graph of y = 10x / (1 + .25x2) for x ≥ 0. Find the coordinates of the maximum point. Figure 5: y y = 10x 1+ 25x²
> Simplify the following expressions. ln 2 - ln x + ln 3
> Differentiate the following functions. y = (x2 ln x) / 2
> Differentiate the following functions. y = ln x / ln 3
> Differentiate the following functions. y = 3 ln x + ln 2
> Human hands covered with cotton fabrics impregnated with the insect repellent DEPA were inserted for 5 minutes into a test chamber containing 200 female mosquitoes. The function f (x) = 26.48 - 14.09 ln x gives the number of mosquito bites received when
> Find the maximum area of a rectangle in the first quadrant with one corner at the origin, an opposite corner on the graph of y = -ln x, and two sides on the coordinate axes.
> Suppose that the total revenue function for a manufacturer is R(x) = 300 ln(x + 1), so the sale of x units of a product brings in about R(x) dollars. Suppose also that the total cost of producing x units is C(x) dollars, where C(x) = 2x. Find the value o
> If the demand equation for a certain commodity is p = 45/(ln x), determine the marginal revenue function for this commodity, and compute the marginal revenue when x = 20.
> The function y = 2x2 - ln 4x (x > 0) has one minimum point. Find its first coordinate.
> The following function may be viewed as a composite function h (x) = f ( g (x)). Find f (x) and g (x). h(x) = 1/x3 - 5x2 + 1
> The graph of the function y = x2 - ln x is shown in Fig. 8. Find the coordinates of its minimum point. Figure 8: 5 4. 3 2 1 Y 0 f(x) = x² - In x 0.5 1.0 1.5
> Use the fact that at the beginning of 1998, the population of the United States was 268,924,000 people and growing at the rate of 1,856,000 people per year. At the beginning of 1998, the annual consumption of ice cream in the United States was 12,582,000
> Repeat Exercise 31 with the functions y = x + ln x and y = ln 5x. (See Fig. 7). Figure 7: Exercise 31: The graphs of y = x + ln x and y = ln 2x are shown in Fig. 6. (a) Show that both functions are increasing for x > 0. (b) Find the point of inters
> The graphs of y = x + ln x and y = ln 2x are shown in Fig. 6. (a) Show that both functions are increasing for x > 0. (b) Find the point of intersection of the graphs. Figure 6: 3 CO 2 1 -1 -2 -3 y = x + Inx 1.0 y = In 2r + 1.5 2.0
> Repeat the previous exercise with y = √x ln x. Exercise 29: Find the coordinates of the relative extreme point of y = x2 ln x, x > 0. Then, use the second derivative test to decide if the point is a relative maximum point or a relative minimum point.
> Find the coordinates of the relative extreme point of y = x2 ln x, x > 0. Then, use the second derivative test to decide if the point is a relative maximum point or a relative minimum point.
> Find the equations of the tangent lines to the graph of y = ln |x| at x = 1 and x = -1.
> Determine the domain of definition of the given function. (a) f (t) = ln(ln t) (b) f (t) = ln(ln(ln t))
> The function f (x) = (ln x + 1)>x has a relative extreme point for x > 0. Find the coordinates of the point. Is it a relative maximum point?
> Write the equation of the tangent line to the graph of y = ln(x2 + e) at x = 0.
> The following function may be viewed as a composite function h (x) = f ( g (x)). Find f (x) and g (x). h(x) = (5x2 + 1)-1/2
> The graph of f (x) = x/(ln x + x) is shown in Fig. 5. Find the coordinates of the minimum point. Figure 5: 1 Y 10 x
> The graph of f (x) = (ln x)/√x is shown in Fig. 4. Find the coordinates of the maximum point. Figure 4: 1 Y 25
> Use the fact that at the beginning of 1998, the population of the United States was 268,924,000 people and growing at the rate of 1,856,000 people per year. At the beginning of 1998, the annual per capita consumption of gasoline in the United States was
> Find the second derivatives. d2/dt2 ln(ln t)
> Find the second derivatives. d2/dt2 (t2 ln t)
> Differentiate the following functions. y = ln (x2 + 1) /(x2 + 1)
> Differentiate the following functions. y = (x3 + 1) ln(x3 + 1)
> Differentiate the following functions. y = (ln x)2
> Differentiate the following functions. y = ln x / ln 2x
> Differentiate the following functions. y = ln x ln 2x
> The following function may be viewed as a composite function h (x) = f ( g (x)). Find f (x) and g (x). h(x) = √(4 - x2)
> Differentiate the following functions. y = 1/ln x
> Differentiate the following functions. y = ln(ex + e-x)
> Differentiate the following functions. y = ln(3x4 - x2)
> A manufacturer plans to decrease the amount of sulfur dioxide escaping from its smokestacks. The estimated cost–benefit function is where f (x) is the cost in millions of dollars for eliminating x% of the total sulfur dioxide. (See Fi
> Differentiate the following functions. y = ln (1/x2)
> Differentiate the following functions. y = ln √x
> Differentiate the following functions. y = ln x2
> Differentiate the following functions. y = 1/(2 + 3 ln x)
> Differentiate the following functions. y = ln x / √x
> The following function may be viewed as a composite function h (x) = f ( g (x)). Find f (x) and g (x). h(x) = (9x2 + 2x - 5)7
> Differentiate the following functions. y = e1+ln x
> Differentiate the following functions. y = ex ln x
> Differentiate the following functions. y = 3 ln x / x
> Find ln (1/e2).
> The width of a rectangle is increasing at a rate of 3 inches per second and its length is increasing at the rate of 4 inches per second. At what rate is the area of the rectangle increasing when its width is 5 inches and its length is 6 inches?
> Find ln(√e).
> A news item is spread by word of mouth to a potential audience of 10,000 people. After t days, f (t) = 10,000 / (1 + 50e-0.4t) people will have heard the news. The graph of this function is shown in Fig. 7. (a) Approximately how many people will have h
> Examine formula (8) for the amount A(t) of excess glucose in the bloodstream of a patient at time t. Describe what would happen if the rate r of infusion of glucose were doubled.
> When a grand jury indicted the mayor of a certain town for accepting bribes, the newspaper, online news outlets, radio, and television immediately began to publicize the news. Within an hour, one-quarter of the citizens heard about the indictment. Estima
> A student learns a certain amount of material for some class. Let f (t) denote the percentage of the material that the student can recall t weeks later. The psychologist Hermann Ebbinghaus found that this percentage of retention can be modeled by a funct
> The following function may be viewed as a composite function h (x) = f ( g (x)). Find f (x) and g (x). h(x) = (x3 + 8x - 2)5
> If f (x) = 3(1 - e-10x), show that y = f (x) satisfies the differential equation y ’ = 10(3 - y), f (0) = 0.
> If y = 5(1 - e-2x), compute y and show that y ’ = 10 - 2y.
> If y = 2(1 - e-x), compute y and show that y ’ = 2 - y.
> Consider the function g(x) = 10 - 10e-0.1x, x ≥ 0. (a) Show that g(x) is increasing and concave down for x ≥ 0. (b) Explain why g(x) approaches 10 as x gets large. (c) Sketch the graph of g(x), x ≥ 0.
> A model incorporating growth restrictions for the number of bacteria in a culture after t days is given by f (t) = 5000(20 + te-0.04t). (a) Graph f (t) and f ‘(t) in the window [0, 100] by [-700, 300]. (b) How fast is the culture changing after 100 days
> Let s(t) be the number of miles a car travels in t hours. Then, the average velocity during the first t hours is / miles per hour. If the average velocity is maximized at time t0, show that at this time the average velocity / equals the instantaneous vel
> Differentiate the functions. y = x(x2 + 1)4
> After a drug is taken orally, the amount of the drug in the bloodstream after t hours is f (t) = 122(e-0.2t - e-t) units. (a) Graph f (t), f (t), and f ‘(t) in the window [0, 12] by [-20, 75] (b) How many units of the drug are in the bloodstream after 7
> Describe an experiment that a doctor could perform to determine the velocity constant of elimination of glucose for a particular patient.
> A news item is broadcast by mass media to a potential audience of 50,000 people. After t days, f (t) = 50,000(1 - e-0.3t) people will have heard the news. The graph of this function is shown in Fig. 8. (a) How many people will have heard the news after
> Compute f (g (x)), where f (x) and g (x) are the following: f (x) = (x + 1)/(x – 3), g (x) = x + 3
> Differentiate the functions. y = x / (x + 1/x)
> Physiologists usually describe the continuous intravenous infusion of glucose in terms of the excess concentration of glucose, C(t) = A(t)/V, where V is the total volume of blood in the patient. In this case, the rate of increase in the concentration of
> Consider the function f (x) = 5(1 - e-2x), x ≥ 0. (a) Show that f (x) is increasing and concave down for all x ≥ 0. (b) Explain why f (x) approaches 5 as x gets large. (c) Sketch the graph of f (x), x ≥ 0.
> Graph y = ln 5x and y = 2 together and determine the x-coordinate of their point of intersection (to four decimal places). Express this number in terms of a power of e.
> Graph y = e2x and y = 5 together, and determine the x-coordinate of their point of intersection (to four decimal places). Express this number in terms of a logarithm.
> Graph the function y = ln(ex), and use trace to convince yourself that it is the same as the function y = x. What do you observe about the graph of y = eln x?
> Find k such that 2-x/5 = ekx for all x.
> Find k such that 2x = ekx for all x.
> Let R(x) be the revenue received from the sale of x units of a product. The average revenue per unit is defined by AR = R(x)/x. Show that at the level of production where the average revenue is maximized, the average revenue equals the marginal revenue.
> Under certain geographic conditions, the wind velocity υ at a height x centimeters above the ground is given by υ = K ln(x/x0), where K is a positive constant (depending on the air density, average wind velocity, and the like) and x0 is a roughness param
> When a drug or vitamin is administered intramuscularly (into a muscle), the concentration in the blood at time t after injection can be approximated by a function of the form The graph of f (t) = 5(e- 0.01t - e- 0.51t), for t ≥ 0, is
> Compute f (g (x)), where f (x) and g (x) are the following: f (x) = x (x2 + 1), g (x) = √x
> Solve for t. 4e0.01t - 3e0.04t = 0
> Solve for t. e0.05t - 4e-0.06t = 0
> Find the coordinates of each relative extreme point of the given function, and determine if the point is a relative maximum point or a relative minimum point. f (x) = 5x - 2ex
> Find the coordinates of each relative extreme point of the given function, and determine if the point is a relative maximum point or a relative minimum point. f (x) = e-x + 3x
> Find the x-intercepts of y = (x - 1)2 ln(x + 1), x > -1.
> (a) Find the point on the graph of y = e-x where the tangent line has slope -2. (b) Plot the graphs of y = e-x and the tangent line in part (a).
> The graph of f (x) = -1 + (x - 1)2ex is shown in Fig. 5. Find the coordinates of the relative maximum and minimum points. Figure 5: f(x) = 1 + (x-1)² ex -2 0 Y 2 - X
> (a) Find the first coordinates of the points on the graph in Fig. 4 where the tangent line has slope 3. (b) Are there any points on the graph where the tangent line has slope -7? Explain. Figure 4: 1 0 Y f(x) = -5x + ex 1 2 x
> A cigar manufacturer produces x cases of cigars per day at a daily cost of 50x(x + 200)/(x + 100) dollars. Show that his cost increases and his average cost decreases as the output x increases.
> Use the second derivative to show that the graph in Fig. 4 is always concave up. Figure 4: 1 0 Y f(x) = -5x + ex 1 2 x
> Determine the growth constant k, then find all solutions of the given differential equation. y/3 = 4y ’
> The graph of f (x) = -5x + ex is shown in Fig. 4. Find the coordinates of the minimum point. Figure 4: 1 0 Y f(x) = -5x + ex 1 2 x
