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mathematics
applied calculus
Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions
Write each expression in Exercises in the form 2kx or 3kx, for a suitable constant k. (3x. 3x/5)5, (16¹/4.16-3/4)3x
Determine the intervals where the function f(x) = ln(x2 + 1) is increasing and where it is decreasing.
Give an example of the use of logarithmic differentiation.
Evaluate the given expressions. Use ln 2 = .69 and ln 3 = 1.1.(a) ln 4 (b) ln 6 (c) ln 54
Find a number b such that the function f(x) = 3-2x can be written in the form f(x) = bx.
Which of the following is the same as(a) ln 4x (b) 4x(c) ln 8x2 - ln 2x (d) none of these In 8x² -? In 2x
Evaluate the given expressions. Use ln 2 = .69 and ln 3 = 1.1.(a) ln 12 (b) ln 16 (c) ln(9 · 24)
Find b so that 8-x/3 = bx for all x.
Evaluate the given expressions. Use ln 2 = .69 and ln 3 = 1.1.(a) ln 1/6(b) ln 2/9(c) ln 1/√2
Evaluate the given expressions. Use ln 2 = .69 and ln 3 = 1.1.(a) ln 100 - 2 ln 5 (b) ln 10 + ln 1/5(c) ln √108
Find the second derivatives. d² dt z (1² In 1)
Find the second derivatives. d² dt In(In t)
Find an equation of the tangent line to the graph of f (x) = ex, where x = -1.
Estimate the slope of ex at x = 0 by calculating the slopeof the secant line passing through the points (0, 1) and (h, eh). Take h = .01, .001, and .0001. - yo h 1
Which of the following is the same as 4 ln 2x?(a) ln 8x (b) 8 ln x(c) ln 8 + ln x (d) ln 16x4
Find the point on the graph of f(x) = ex, where the tangent line is parallel to y = x.
Which of the following is the same as ln(9x) - ln(3x)?(a) ln 6x (b) ln(9x)/ ln(3x)(c) 6 · ln(x) (d) ln 3
The graph of f (x) = (ln x)/√x is shown in Fig. 4. Find the coordinates of the maximum point. 1 y Figure 4 + 25
Simplify the function before differentiating.f (x) = (e3x)5
Simplify the function before differentiating. f(x) = 1 Vet
Which of the following is the same as ln 9x2?(a) 2 · ln 9x (b) 3x · ln 3x(c) 2 · ln 3x (d) none of these
Simplify the function before differentiating.f(x) = exe2xe3x
Suppose that A = (a, b) is a point on the graph of ex. What is the slope of the graph of ex at the point A?
Solve the given equation for x.ln x - ln x2 + ln 3 = 0
The graph of f (x) = x/(ln x + x) is shown in Fig. 5. Find the coordinates of the minimum point. 1 y Figure 5 10 X
Simplify the function before differentiating. f(x)= = et + 5e2x ex
Find the slope–point form of the equation of the tangent line to the graph of ex at the point (a, ea).
Solve the given equation for x.ln √x - 2 ln 3 = 0
Simplify the function before differentiating.f (t) = e3t(e2t - e4t)
The graph of the functions f(x) = ex2 - 4x2 is shown in Fig. 1. Find the first coordinates of the relative extreme points. f(x) = ex² - 4x² -1.5 Figure 1 - -1/-0.5 12 10 8 6 4 2 y + 0.5 1 1.5 X
Simplify the function before differentiating. f(t) = √₂³x Ve3x
Solve the given equation for x.ln x4 - 2 ln x = 1
Show that the function in Fig. 1 has a relative maximum at x = 0 by determining the concavity of the graph at x = 0. f(x) = ex² - 4x² -1.5 -1/-0.5 Figure 1 12 10 8 6 4 2 y + 0.5 1 1.5
Write the equation of the tangent line to the graph of y = ln(x2 + e) at x = 0.
Solve the given equation for x.ln x2 - ln 2x + 1 = 0
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?
Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points. f(x): 3 - 4x e²x
Solve the given equation for x.(ln x)2 - 1 = 0
Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points.f(x) = (1 + x)e-3x
Determine the domain of definition of the given function.(a) f (t) = ln(ln t) (b) f (t) = ln(ln(ln t))
Solve the given equation for x.3 ln x - ln 3x = 0
Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points.f (x) = (1 - x)e2x
Find the equations of the tangent lines to the graph of y = ln |x| at x = 1 and x = -1.
Find the values of x at which the function has a possible relative maximum or minimum point. (Recall that ex is positive for all x.) Use the second derivative to determine the nature of the function at these points. f(x) = = 4x - 1 e-x/2
Solve the given equation for x.ln √x = √ln x
Solve the following equations for t.4e0.03t - 2e0.06t = 0
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.
Solve the given equation for x.2( ln x)2 + ln x - 1 = 0
Solve the following equations for t.et - 8e0.02t = 0
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.
Find the extreme points on the graph of y = x2ex, and decide which one is a maximum and which one is a minimum.
Differentiate.y = ln[e2x(x3 + 1)(x4 + 5x)]
A painting purchased in 2015 for $100,000 is estimated to be worth v(t) = 100,000et/5 dollars after t years. At what rate will the painting be appreciating in 2020?
Determine the intervals where the function f (x) = x ln x (x > 0) is increasing and where it is decreasing.
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 of x at which the profit function R(x) - C(x)
Find the point on the graph of y = (1 + x2)ex where the tangent line is horizontal.
The velocity of a parachutist during free fall is f (t) = 60(1 - e-0.17t) meters per second. Answer the following questions by reading the graph in Fig. 2. (Recall that acceleration is the derivative of velocity.)(a) What is the velocity when t = 8 seconds?(b) What is the acceleration when t =
Differentiate.y = ln [√xex² +1]
The expressions in Exercises may be factored as shown. Find the missing factors. 23th = 2³(
Suppose that the velocity of a parachutist is v(t) = 65(1 - e-0.16t) meters per second. The graph of v(t) is similar to that in Fig. 2. Calculate the parachutist’s velocity and acceleration when t = 9 seconds. 60 50 40 30 20 10 y Figure 2 y = f(t) y = f'(t) 10
The expressions in Exercises may be factored as shown. Find the missing factors. 52+h = 25(
Show that the tangent lines to the graph of x = 1 and x = -1 are parallel. y et - ex et + ex at
The height of a certain plant, in inches, after t weeks isThe graph of f (t) resembles the graph in Fig. 3. Calculate the rate of growth of the plant after 7 weeks. f(t) = 1 .05 + e-0.4t*
The graph of f (x) = -5x + ex is shown in Fig. 4. Find the coordinates of the minimum point. 1 0 Y Figure 4 f(x) = 5x + ex 1 2 X
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.
Show that the tangent line to the graph of y = ex at the point (a, ea) is perpendicular to the tangent line to the graph of y = e-x at the point (a, e-a).
Differentiate.y = 1n (x + 1/ x - 1)
Use the second derivative to show that the graph in Fig. 4 is always concave up. 1 0 Y Figure 4 f(x) = 5x + ex 1 2 X
The expressions in Exercises may be factored as shown. Find the missing factors. 2x+h-2x = 2*(
The expressions in Exercises may be factored as shown. Find the missing factors. 5x+h+5^ = 5 (
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 the concentration was x percent.(a) Graph f
Find the slope of the tangent line to the curve y = xex at (0, 0).
Differentiate.y = ln (x + 1)4 /ex-1
(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. 1 0 Y Figure 4 f(x) = 5x + ex 1 2 X
The expressions in Exercises may be factored as shown. Find the missing factors. 3x/2 + 3-x/2 = 3-x/2(
Find the slope of the tangent line to the curve y = xex at (1, e).
Differentiate.y = ln(3x + 1) ln(5x + 1)
The graph of f (x) = -1 + (x - 1)2ex is shown in Fig. 5. Find the coordinates of the relative maximum and minimum points. f(x) = 1 + (x-1)² ex Figure 5 1 -2 0 -1 y + 2 X
Let a and b be positive numbers. A curve whose equation is y = e-ae-bx is called a Gompertz growth curve. These curves are used in biology to describe certain types of population growth. Compute the derivative of y = e-2e-0.01x.
The expressions in Exercises may be factored as shown. Find the missing factors. 57x/2 - 5x/2 = √√5x(
Differentiate.y = ( ln 4x)( ln 2x)
Find dy/dx if y = e-(1/10)e-x/2.
Use logarithmic differentiation to differentiate the following functions.f(x) = (x + 1)4 (4x - 1)2
Use logarithmic differentiation to differentiate the following functions. f(x) = (x + 1)(2x + 1)(3x + 1) √4x + 1
Let f(t) be the function from Exercise 39 that gives the height (inches) of a plant at time t (weeks).(a) When is the plant 11 inches tall?(b) When is the plant growing at the rate of 1 inch per week?(c) What is the fastest rate of growth of the plant, and when does this occur?Exercise 39The height
In a study, a cancerous tumor was found to have a volume of f(t) = 1.8253(1 - 1.6e-0.4196t)3 milliliters after t weeks, with t > 1.(a) Sketch the graphs of f (t) and f′(t) for 1 ≤ t ≤ 15. What do you notice about the tumor’s volume?(b) How large is the tumor after 5 weeks?(c) When will
(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).
(a) Use the fact that e4x = (ex)4 to findSimplify the derivative as much as possible.(b) Take an approach similar to the one in (a) and show that, if k is a constant, d dx -(e4x).
Use logarithmic differentiation to differentiate the following functions. f(x)= (x - 2)³(x-3)4 (x + 4)5
Graph the function f(x) = 2x in the window [-1, 2] by [-1, 4], and estimate the slope of the graph at x = 0.
Use logarithmic differentiation to differentiate the following functions.f(x) = ex(3x - 4)8
Find the x-intercepts of y = (x - 1)2 ln(x + 1), x > -1.
Graph the function f(x) = 3x in the window [-1, 2] by [-1, 8], and estimate the slope of the graph at x = 0.
In Exercises, 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
Solve the following equations for t.t ln t = e
By trial and error, find a number of the form b = 2. (just one decimal place) with the property that the slope of the graph of y = bx at x = 0 is as close to 1 as possible.
In Exercises, 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
Solve the following equations for t.ln( ln 3t) = 0
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 f(t) = c(e-k1t - e-k2t). The graph of f (t) = 5(e- 0.01t - e- 0.51t), for t ≥ 0, is shown in Fig. 6. Find the value of t
Sketch the graphs of the following functions.y = e2x
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