Solve for t. e0.05t - 4e-0.06t = 0
> 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 (1/x)
> 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
> 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
> Solve the following equations for x. (ex)2 * e2-3x = 4
> Solve the following equations for x. 4ex * e-2x = 6
> Solve the following equations for x. e5x * eln 5 = 2
> Solve the following equations for x. (e2)x * eln 1 = 4
> Solve the following equations for x. 750e-0.4x = 375
> Solve the following equations for x. 5 ln 2x = 8
> Solve the following equations for x. e√x = √(ex)
> Solve the following equations for x. 2ex/3 - 9 = 0
> A sugar refinery can produce x tons of sugar er week at a weekly cost of .1x2 + 5x + 2250 dollars. Find the level of production for which the average cost is at a minimum and show that the average cost equals the marginal cost at that level of production
> Differentiate the function. y = 1/ π + 2 / x2 + 1
> Solve the following equations for x. 2 ln x = 7
> Solve the following equations for x. ln(ln 3x) = 0
> Solve the following equations for x. ln(x2 - 5) = 0
> Solve the following equations for x. ln 3x = ln 5
> Solve the following equations for x. 4 - ln x = 0
> Solve the following equations for x. 6e-0.00012x = 3
> Solve the following equations for x. कर = 25 =
> Solve the following equations for x. ln x2 = 9
> Solve the following equations for x. ln 3x = 2
> Solve the following equations for x. ln(4 - x) = 12
> Determine the growth constant k, then find all solutions of the given differential equation. y = 1.6y’
> A closed rectangular box is to be constructed with one side 1 meter long. The material for the top costs $20 per square meter, and the material for the sides and bottom costs $10 per square meter. Find the dimensions of the box with the largest possible
> Solve the following equations for x. e1-3x = 4
> Solve the following equations for x. e2x = 5
> Simplify the following expressions. eln 3-2 ln x
> Simplify the following expressions. eln x+ln 2
> Simplify the following expressions. ln(e-2e4)
> Simplify the following expressions. e-2 ln 7
> Simplify the following expressions. ex ln 2
> Simplify the following expressions. e2 ln x
> Simplify the following expressions. e4 ln 1
> Determine the growth constant k, then find all solutions of the given differential equation. 2 y' – y/2 = 0
> Simplify the following expressions. ln(ln e)
> An open rectangular box is 3 feet long and has a surface area of 16 square feet. Find the dimensions of the box for which the volume is as large as possible.
> Simplify the following expressions. ln (e-2ln e)
> Simplify the following expressions. Celh 1
> Simplify the following expressions. eln 4.1
> Simplify the following expressions. ln e-3
> If ln x = 4.5, write x using the exponential function.
> If ln x = -1, write x using the exponential function.
> If e-x = 3.2, write x in terms of the natural logarithm.
