Simplify the following expressions. e2 ln x
> 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
> 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. 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.
> If ex = 5, write x in terms of the natural logarithm.
> Determine the growth constant k, then find all solutions of the given differential equation. y' - 6y = 0
> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. y5 - 3x2 = x
> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x3 + y3 - 6 = 0
> A function h (x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = [f (x)/x2]
> Suppose that x and y are related by the given equation and use implicit differentiation to determine dy/dx. x2 - y2 = 1
> The annual sales S (in dollars) of a company may be approximated by the formula S = 50,000√(e√t), where t is the number of years beyond some fixed reference date. Use a logarithmic derivative to determine the percentage rate of growth of sales at t = 4
> Find the logarithmic derivative and then determine the percentage rate of change of the function at the point indicated. g(p) = 5/(2p + 3) at p = 1 and p = 11
> Find the logarithmic derivative and then determine the percentage rate of change of the function at the point indicated. f (p) = 1/(p + 2) at p = 2 and p = 8
> Find the logarithmic derivative and then determine the percentage rate of change of the function at the point indicated. G(s) = e-0.05s2 at s = 1 and s = 10
> Find the logarithmic derivative and then determine the percentage rate of change of the function at the point indicated. f (t) = e0.3t2 at t = 1 and t = 5
> Find the logarithmic derivative and then determine the percentage rate of change of the function at the point indicated. f (x) = e-0.05x at x = 1 and x = 10
> Consider the exponential decay function y = P0e-λt, with time constant T. We define the time to finish to be the time it takes for the function to decay to about 1% of its initial value P0. Show that the time to finish is about four times the time consta
> Find the logarithmic derivative and then determine the percentage rate of change of the function at the point indicated. f (x) = e0.3x at x = 10 and x = 20
> Consider the demand function q = 60,000e-0.5p from Check Your Understanding 5.3. (a) Determine the value of p for which the value of E(p) is 1. For what values of p is demand inelastic? (b) Graph the revenue function in the window [0, 4] by [-5000, 50,00
> A cost function C(x) gives the total cost of producing x units of a product. The elasticity of cost at quantity x, Ec (x), is defined to be the ratio of the relative rate of change of cost (with respect to x) divided by the relative rate of change of qua
> A function h (x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = f (x) / (x2 + 1)
> A cost function C(x) gives the total cost of producing x units of a product. The elasticity of cost at quantity x, Ec (x), is defined to be the ratio of the relative rate of change of cost (with respect to x) divided by the relative rate of change of qua
> A cost function C(x) gives the total cost of producing x units of a product. The elasticity of cost at quantity x, Ec (x), is defined to be the ratio of the relative rate of change of cost (with respect to x) divided by the relative rate of change of qua
> A cost function C(x) gives the total cost of producing x units of a product. The elasticity of cost at quantity x, Ec (x), is defined to be the ratio of the relative rate of change of cost (with respect to x) divided by the relative rate of change of qua
> Show that any demand function of the form q = a/pm has constant elasticity m.
> A country that is the major supplier of a certain commodity wishes to improve its balance-of-trade position by lowering the price of the commodity. The demand function is q = 1000/p2. (a) Compute E(p). (b) Will the country succeed in raising its revenue?
> A subway charges 65 cents per person and has 10,000 riders each day. The demand function for the subway is q = 2000 √(90 – p). (a) Is demand elastic or inelastic at p = 65? (b) Should the price of a ride be raised or lowered to increase the amount of mon
> Suppose that the function P(t) satisfies the differential equation y ’(t) = -.5y(t), y(0) = 10. (a) Find an equation of the tangent line to the graph of y = P(t) at t = 0. (b) Find P(t). (c) What is the time constant of the decay curve y = P(t)?
> A movie theater has a seating capacity of 3000 people. The number of people attending a show at price p dollars per ticket is q = (18,000/p) - 1500. Currently, the price is $6 per ticket. (a) Is demand elastic or inelastic at p = 6? (b) If the price is l
> An electronic store can sell q = 10,000/(p + 50) - 30 cellular phones at a price p dollars per phone. The current price is $150. (a) Is demand elastic or inelastic at p = 150? (b) If the price is lowered slightly, will revenue increase or decrease?
> Find the logarithmic derivative and then determine the percentage rate of change of the function at the point indicated. f (t) = t10 at t = 10 and t = 50
> Currently, 1800 people ride a certain commuter train each day and pay $4 for a ticket. The number of people q willing to ride the train at price p is q = 600(5 - 1p). The railroad would like to increase its revenue. (a) Is demand elastic or inelastic at
> A function h (x) is defined in terms of a differentiable f (x). Find an expression for h (x). h(x) = (x2 + 2x - 1) f (x)
> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = 700/(p + 5), p = 15
> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = p2e-(p+3), p = 4
> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = (77/p2) + 3, p = 1
> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = 400(116 - p2), p = 6
> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = 600e-0.2p, p = 10
> Consider an exponential decay function P(t) = P0e-λt, and let T denote its time constant. Show that, at t = T, the function P(t) decays to about one-third of its initial size. Conclude that the time constant is always larger than the half-life.
> For the demand function, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price. q = 700 - 5p, p = 80
> The wholesale price in dollars of one pound of pork is modeled by the function f (t) = 1.4 + .26t - .1t2 + .01t3, where t is measured in years from January 1, 2010. (a) Estimate the price in 2012 and find the percentage rate of increase of the price in 2
> The wholesale price in dollars of one pound of ground beef is modeled by the function f (t) = 3.08 + .57t - .1t2 + .01t3, where t is measured in years from January 1, 2010. (a) Estimate the price in 2011 and find the rate in dollars per year at which the
> The price of wheat per bushel at time t (in months) is approximated by f (t) = 4 + .001t + .01e-t. What is the percentage rate of change of f (t) at t = 0? t = 1? t = 2?
> Find the logarithmic derivative and then determine the percentage rate of change of the function at the point indicated. f (t) = t2 at t = 10 and t = 50
