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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Suppose that ƒ is a function such that ƒ(0) = 1 and for all x, ƒ'(x) = ƒ(x) and ƒ(x) > 0 (in Chapter 7, we will see that ƒ(x) is the exponential function ex). Prove that for all x ≥ 0 (each assertion follows from the previous one),Then prove by induction that for every whole number n and
Sketch the graph of the function. Indicate the transition points and asymptotes. y = 1 x² 1 (x - 2)²
Sketch the graph of the function. Indicate the transition points and asymptotes. y = 4 x²-9
Assume that ƒ" exists and ƒ"(x) = 0 for all x. Prove that ƒ(x) = mx + b, where m = ƒ'(0) and b = ƒ(0).
Sketch the graph of the function. Indicate the transition points and asymptotes. y = 1 (x² + 1)²
Define ƒ(x) = x3 sin(1/x) for x ≠ 0 and ƒ(0) = 0.(a) Show that ƒ' is continuous at x = 0 and that x = 0 is a critical point of ƒ.(b) Examine the graphs of ƒ and ƒ'. Can the First Derivative Test be applied?(c) Show that ƒ(0) is neither a local min nor a local max.
Sketch the graph of the function. Indicate the transition points and asymptotes. y = x² (x² - 1)(x² + 1)
Suppose that ƒ(x) satisfies the following equation (an example of a differential equation):(a) Show that ƒ(x)2 + ƒ'(x)2 = ƒ(0)2 + ƒ'(0)2 for all x. Show that the function on the left has zero derivative.(b) Verify that sin x and cos x satisfy Eq. (1), and deduce that sin2 x + cos2 x =
Sketch the graph of the function. Indicate the transition points and asymptotes. y = 1 √x² + 1
Suppose that functions ƒ and g satisfy Eq. (1) and have the same initial values—that is, ƒ(0) = g(0) and ƒ'(0) = g(0). Prove that ƒ(x) = g(x) for all x. Apply Exercise 70(a) to ƒ − g.Eq.(1)Data From Exercise 70Suppose that ƒ(x) satisfies the following equation (an example of a
Sketch the graph of the function. Indicate the transition points and asymptotes. y = √x² + 1
Use Exercise 71 to prove ƒ(x) = sin x is the unique solution of Eq. (1) such that ƒ(0) = 0 and ƒ'(0) = 1; and g(x) = cos x is the unique solution such that g(0) = 1 and g'(0) = 0. This result can be used to develop all the properties of the trigonometric functions “analytically”—that is,
We explore functions whose graphs approach a nonhorizontal line as x → ∞. A line y = ax + b is called a slant asymptote if or lim (f(x) (ax+b)) = 0 X-00 lim (f(x) (ax + b)) = 0 X--∞0
We explore functions whose graphs approach a nonhorizontal line as x → ∞. A line y = ax + b is called a slant asymptote ifIf ƒ(x) = P(x)/Q(x), where P and Q are polynomials of degrees m + 1 and m, then by long division, we can writewhere P1 is a polynomial of degree or lim (f(x) (ax+b)) =
We explore functions whose graphs approach a nonhorizontal line as x → ∞. A line y = ax + b is called a slant asymptote ifSketch the graph ofProceed as in the previous exercise to find the slant asymptote.We explore functions whose graphs approach a nonhorizontal line as x → ∞. A line y =
We explore functions whose graphs approach a nonhorizontal line as x → ∞. A line y = ax + b is called a slant asymptote ifShow that y = 3x is a slant asymptote for ƒ(x) = 3x + x−2. Determine whether ƒ(x) approaches the slant asymptote from above or below, and make a sketch of the graph.
We explore functions whose graphs approach a nonhorizontal line as x → ∞. A line y = ax + b is called a slant asymptote ifSketch the graph of or lim (f(x) (ax+b)) = 0 X-00 lim (f(x) (ax + b)) = 0 X--∞0
Assume that ƒ' and ƒ" exist for all x and let c be a critical point of ƒ. Show that ƒ(x) cannot make a transition from ++ to −+ at x = c. Apply the MVT to ƒ'(x).
Assume that ƒ" exists and ƒ"(x) > 0 for all x. Show that ƒ(x) cannot be negative for all x. Show that ƒ'(b) ≠ 0 for some b and use the result of Exercise 62 in Section 4.4.Data From Exercise 62 From Section 4.4Prove that if ƒ" exists and ƒ"(x) > 0 for all x, then the graph of ƒ“
Fill in a table of the following type:ƒ(u) = tan u, g(x) = x4 f(g(x)) f'(u) f'(g(x)) g'(x) (fog)'(x)
Discuss how it is possible to be speeding up with a velocity that is decreasing.
Sketch a graph of position as a function of time for an object that is speeding up and has negative acceleration.
On an exam, Jason was asked to differentiate the equation x2 + 2xy + y3 = 7Find the errors in Jason’s answer: 2x + 2xy'+ 3y2 = 0.
Differentiate the expression with respect to x, assuming that y is implicitly a function of x.x2y3
Calculate y" and y"'.y = x4 − 25x2 + 2x
For each, give an equation of the tangent line to the graph at x = 0.(a) y = sin x(b) y = cos x
Find an equation of the tangent line at the point indicated.y = tan x, x = π/4
Assign variables and restate the following problem in terms of known and unknown derivatives (but do not solve it): How fast is the volume of a cube increasing if its side increases at a rate of 0.5 cm/s?
The radius of a circular oil slick expands at a rate of 2 m/min.(a) How fast is the area of the oil slick increasing when the radius is 25 m?(b) If the radius is 0 at time t = 0, how fast is the area increasing after 3 min?
At what rate is the diagonal of a cube increasing if its edges are increasing at a rate of 2 cm/s?
Refer to the function ƒ whose graph is shown in Figure I.Estimate ƒ'(0.7) and ƒ'(1.1). 6- 5432 5- 0.5 1.0 1.5 2.0 -X
State whether each claim is true or false. If false, give an example demonstrating that it is false.(a) If ƒ is continuous at a, then ƒ is differentiable at a.(b) If ƒ is differentiable at a, then ƒ is continuous at a.
Suppose that ƒ'(4) = g(4) = g'(4) = 1. Do we have enough information to compute F'(4), where F(x) = ƒ(g(x))? If not, what is missing?
Write the function as a composite ƒ(g(x)) and compute the derivative using the Chain Rule.y = (x + sin x)4
Compute the derivative.ƒ(x) = sin x cos x
Water pours into a cylindrical glass of radius 4 cm. Let V and h denote the volume and water level, respectively, at time t.Restate this question in terms of dV/dt and dh/dt: How fast is the water level rising if water pours in at a rate of 2 cm3/min?
What is the millionth derivative of ƒ(x) = ex?
Compute ƒ'(a) using the limit definition and find an equation of the tangent line to the graph of ƒ at x = a.ƒ(x) = 5 − 3x, a = 2
Compute ƒ'(a) using the limit definition and find an equation of the tangent line to the graph of ƒ at x = a.ƒ(x) = x−1, a = 4
Compute ƒ'(a) using the limit definition and find an equation of the tangent line to the graph of ƒ at x = a.ƒ(x) = x3, a = −2
Compute dy/dx using the limit definition.y = 4 − x2
Compute dy/dx using the limit definition.y = √2x + 1
Differentiate the expression with respect to x, assuming that y is implicitly a function of x.sin y/x
Compute the derivative.h(t) = 9 csc t + t cot t
Calculate y" and y"'.y = 5t−3 + 7t−8/3
Exercises 14–16 refer to the four graphs of s as a function of t in Figure 7.Sketch s' for each of the four graphs of s. y A v C B ·x
Calculate y" and y"'.y = (x2 + x)(x3 + 1)
Compute the derivative. y = √√√x + √√x + √x
A planet’s period P (number of days to complete one revolution around the sun) is approximately 0.199A3/2, where A is the average distance (in millions of kilometers) from the planet to the sun. Calculate P and dP/dA for Earth using the value A = 150.
A roller coaster has the shape of the graph in Figure 20. Show that when the roller coaster passes the point (x, ƒ (x)), the vertical velocity of the roller coaster is equal to ƒ'(x) times its horizontal velocity. (x, f(x))
In Exercises 41–44, Calculate ƒ'(x) in terms of P(x), Q(x), and R(x), assuming that P'(x) = Q(x), Q'(x) = −R(x), and R'(x) = P(x).ƒ(x) = xR(x) + Q(x)
Find the points on the graph of y2 = x3 − 3x + 1 (Figure 5) where the tangent line is horizontal.(a) First show that 2yy'= 3x2 − 3, where y'= dy/dx.(b) Do not solve for y'. Rather, set y'= 0 and solve for x. This yields two values of x where the slope may be zero.(c) Show that the positive
Two trains leave a station at t = 0 and travel with constant velocity v along straight tracks that make an angle θ.(a) Show that the trains are separating from each other at a rate v √2 − 2 cos θ.(b) What does this formula give for θ = π?
Use the Chain Rule to find the derivative.y =√cot9 θ + 1
As the wheel of radius r cm in Figure 21 rotates, the rod of length L attached at point P drives a piston back and forth in a straight line. Let x be the distance from the origin to point Q at the end of the rod, as shown in the figure.(a) Use the Pythagorean Theorem to show that(b) Differentiate
Let y = tan x. Then y'= sec2 x and by the Product Rule,y", y"', y = t2 sin t
Compute the derivative.h(z) = (z + (z + 1)1/2)−3/2
In Exercises 41–44, Calculate ƒ'(x) in terms of P(x), Q(x), and R(x), assuming that P'(x) = Q(x), Q'(x) = −R(x), and R'(x) = P(x).ƒ(x) = Q(x)P(x)
Show, by differentiating the equation, that if the tangent line at a point (x, y) on the curve x2y − 2x + 8y = 2 is horizontal, then xy = 1. Then substitute y = x−1 in x2y − 2x + 8y = 2 to show that the tangent line is horizontal at the points (2, 1/2) and (− 4, −1/4).
Use the Chain Rule to find the derivative.y = csc(9 − 2θ2)
Calculate the first five derivatives of ƒ(x) = cos x. Then determine ƒ(8)(x) and ƒ(37)(x).
In Exercises 41–44, Calculate ƒ'(x) in terms of P(x), Q(x), and R(x), assuming that P'(x) = Q(x), Q'(x) = −R(x), and R'(x) = P(x). f(x) = P(x) Q(x) X
Compute the derivative.y = tan(t−3)
Show that no point on the graph of x2 − 3xy + y2 = 1 has a horizontal tangent line.
A spectator seated 300 m away from the center of a circular track of radius 100 m watches an athlete run laps at a speed of 5 m/s. How fast is the distance between the spectator and athlete changing when the runner is approaching the spectator and the distance between them is 250 m? The diagram for
Use the Chain Rule to find the derivative.y = cot(√θ − 1)
Compute the derivative.y = 4 cos(2 − 3x)
In Exercises 41–44, Calculate ƒ'(x) in terms of P(x), Q(x), and R(x), assuming that P'(x) = Q(x), Q'(x) = −R(x), and R'(x) = P(x). f(x) = Q(x)R(x) P(x)
Calculate the first five derivatives of ƒ(x) = sin x. Then determine ƒ(9)(x) and ƒ(102)(x).
In Exercises 45–48, calculate the derivative using the values:(ƒg)'(4) and (ƒ/g)'(4) f(4) f'(4) 10 -2 g(4) 5 g'(4) -1
Find the derivative using the appropriate rule or combination of rules.y = tan(x2 + 4x)
The curve x3 + y3 = 3xy (Figure 7) was first discussed in 1638 by the French philosopher ma the matician Ren´e Descartes, who called it the folium (meaning “leaf”). Descartes’s scientific colleague Gilles de Roberval called it the jasmine flower. Both men believed incorrectly that the leaf
Compute the derivative.y = sin(2x) cos2 x
Find the derivative using the appropriate rule or combination of rules.y = sin(x2 + 4x)
Let ƒ(x) = sin x. We can compute ƒ(n)(x) as follows: First, express n = 4m + r where m is a whole number and r = 0, 1, 2, or 3. Then determine ƒ(n)(x) from r. Explain how to do the latter step.
In contrast to Exercise 45, the size of a falling lightweight object may be more significant than its mass when taking into account air resistance. One model that takes such an approach for falling raindrops iswhere s(t) is the distance a raindrop has fallen (in meters), D is the raindrop diameter,
Compute the derivative. y = sin
In Exercises 45–48, calculate the derivative using the values:F'(4), where F(x) = x2 ƒ(x) f(4) f'(4) 10 -2 g(4) 5 g'(4) -1
Find a point on the folium x3 + y3 = 3xy other than the origin at which the tangent line is horizontal.
Let ƒ(x) = sin2 x and g(x) = cos2 x.(a) Use an identity and prove ƒ'(x) = −g'(x) without directly computing ƒ'(x) and g'(x).(b) Now verify the result in (a) by directly computing ƒ'(x) and g'(x).
Exercises 47–49: The Lorenz curve y = F(r) is used by economists to study income distribution in a given country (see Figure 15). By definition, F(r) is the fraction of the total income that goes to the bottom rth part of the population, where 0 ≤ r ≤ 1. For example, iƒ F(0.4) = 0.245, then
Find the derivative using the appropriate rule or combination of rules.y = x cos(1 − 3x)
Compute the derivative. y = t 1 + sect
In a manufacturing process, a drill press automatically drills a hole into a sheet metal part on a conveyor. In the drilling operation the drill bit starts at rest directly above the part, descends quickly, drills a hole, and quickly returns to the start position. The maximum vertical speed of the
Assign the labels y = ƒ(x), y = g(x), and y = h(x) to the graphs in Figure 12 in such a way that ƒ'(x) = g(x) and g'(x) = h(x). A (A) (B) X for (C)
In Exercises 45–48, calculate the derivative using the values:G'(4), where G(x) = (g(x))2
Plot x3 + y3 = 3xy + b for several values of b and describe how the graph changes as b → 0. Then compute dy/dx at the point (b1/3, 0). How does this value change as b → ∞? Do your plots confirm this conclusion?
Find the derivative using the appropriate rule or combination of rules.y = sin(x2) cos(x2)
Use a computer algebra system to compute f(k)(x) for k = 1, 2, 3 for the following functions: (a) f(x)= (1+x³)5/3 (b) f(x) = 1-x² 1-5x - 6x²
Find the x-coordinates of the points where the tangent line is horizontal on the trident curve xy = x3 − 5x2 + 2x − 1, so named by Isaac Newton in his treatise on curves published in 1710 (Figure 8).2x3 − 5x2 + 1 = (2x − 1)(x2 − 2x − 1). -2 20 -20 2 4 + 6 8 00 x
Let ƒ(x) = tan2 x and g(x) = sec2x.(a) Use an identity and prove ƒ'(x) = g'(x) without directly computing ƒ'(x) and g'(x).(b) Now verify the result in (a) by directly computing ƒ'(x) and g'(x).
The following table provides values of F(r) for the United States in 2010. Assume that the national average income was A = $66,000.(a) What was the average income in the lowest 40% of households?(b) Show that the average income of the households belonging to the interval [0.4, 0.6] was $48,180.(c)
Compute the derivative.y = z csc(9z + 1)
Find the derivative using the appropriate rule or combination of rules.y = (4t + 9)1/2
Use the rules in Exercise 48 and the Linearity Rules to prove the first part of Theorem 1 in Section 3.1.Data From Exercise 48Prove each of the following using the definition of the derivative.The First-Power Rule: d/dx x = 1The Constant Rule: d/x c = 0 THEOREM 1 Derivative of Linear and Constant
Find the values of x between 0 and 2π where the tangent line to the graph of y = sin x cos x is horizontal.a
Find an equation of the tangent line at each of the four points on the curve (x2 + y2 − 4x)2 = 2(x2 + y2) where x = 1. This curve (Figure 9) is an example of a limac¸on of Pascal, named after the father of the French philosopher Blaise Pascal, who first described it in 1650. 3 y -3+ D 1 3 5
In Exercises 49 and 50, a rectangle’s length L(t) and width W(t) (measured in inches) are varying in time (t, in minutes). Determine A(t) in each case. Is the area increasing or decreasing at that time?At t = 3, we have L(3) = 4, W(3) = 6, L'(3) = −4, and W'(3) = 5.
Use Exercise 47(c) to prove:(a) F'(r) is an increasing function of r.(b) Income is distributed equally (all households have the same income) if and only if F(r) = r for 0 ≤ r ≤ 1.Data From Exercise 47(c)(c) Let 0 ≤ r ≤ 1. A household belongs to the 100rth percentile if its income is greater
Find the derivative using the appropriate rule or combination of rules.y = (z + 1)4 (2z − 1)3

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