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mathematics
applied calculus

Questions and Answers of Applied Calculus

The graph of y = x - ex has one extreme point. Find its coordinates and decide whether it is a maximum or a minimum.
Differentiate.y = ln[(1 + x)2(2 + x)3(3 + x)4]
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
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
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
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
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
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
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
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
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
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 = In (x + 1)4(x³ + 2) / x-1
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
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
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.
(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

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