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mathematics
calculus 4th

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

Plot the graph ƒ(θ) = sec θ + csc θ over [0, 2π] and determine the number of solutions to ƒ'(θ) = 0 in this interval graphically. Then compute ƒ'(θ) and find the solutions.
Compute the derivative.y = sin100 x
Find the derivative at the points where x = 1 on the folium (x2 + y2)2 = 25/4 xy2. See Figure 10. 2 X
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 = 6, we have L(6) = 6, W(6) = 3, L'(6) = 5, and W'(6) = −2.
What is p(99)(x) for p(x) as in Exercise 50?Data From Exercise 50Find the 100th derivative of p(x) = (x + x³ + x²) ¹0 (1 + x²) ¹¹ (x³ + x³ + x²)
Let g(t) = t − sin t.(a) Plot the graph of g with a graphing utility for 0 ≤ t ≤ 4π.(b) Show that the slope of the tangent line is nonnegative. Verify this on your graph.(c) For which values of t in the given range is the tangent line horizontal?
Use the table of values of ƒ to determine which of (A) or (B) in Figure 14 is the graph of ƒ'. Explain. X 0 0.5 1 1.5 2 2.5 3 3.5 4 237 257 268 f(x) 10 55 98 139 177 210 Kn y (A) (B) -X
Calculate F'(0), whereDo not calculate F'(x). Instead, write F(x) = ƒ(x)/g(x) and express F'(0) directly in terms of ƒ(0), ƒ'(0), g(0), g'(0). F(x)= x² + x³+4x² - 7x x43x² + 2x + 1
Compute the derivative.y = cos(x100)
Plot (x2 + y2)2 = 12(x2 − y2) + 2 for x and y between −4 and 4 using a computer algebra system. How many horizontal tangent lines does the curve appear to have? Find the points where these occur.
Find the derivative using the appropriate rule or combination of rules.y = sin(cos(sin x))
Let ƒ(x) = (sin x)/x for x ≠ 0 and ƒ(0) = 1.(a) Plot ƒ on [−3π, 3π].(b) Show that ƒ(c) = 0 if c = tan c. Use the numerical root finder on a computer algebra system to find a good approximation to the smallest positive value c0 such that ƒ'(c0) = 0.(c) Verify that the horizontal line y =
Let R be a variable and r a constant. Compute the derivatives: (a) R dR (b) d dR d (c) r²R³ dR
Use the Product Rule twice to find a formula for (ƒg)" in terms of ƒ and g and their first and second derivatives.
Proceed as in Exercise 51 to calculate F'(0), whereData From Exercise 51Calculate F'(0), whereDo not calculate F'(x). Instead, write F(x) = ƒ(x)/g(x) and express F'(0) directly in terms of ƒ(0), ƒ'(0), g(0), g'(0). F(x) = (1 + x + x4/3 + x5/3) 3x5 + 5x + 5x + 1 8x97x4+1
Compute the derivative.y = cos(cos(cos(θ)))
Calculate dx/dy for the equation y4 + 1 = y2 + x2 and find the points on the graph where the tangent line is vertical.
Find the derivative using the appropriate rule or combination of rules. y = √sin x cos x
Use the Product Rule to find a formula for (ƒg)"' and compare your result with the expansion of (a + b)3. Then try to guess the general formula for (ƒg)(n).
Use the following table of values to calculate the derivative of the given function at x = 2:S (x) = 3 ƒ(x) − 2g(x) X f(x) g(x) 4 2 2 5 4 3 f'(x) -3 -2 g'(x) 9 3
Show that no tangent line to the graph of ƒ(x) = tan x has zero slope. What is the least slope of a tangent line? Justify by sketching the graph of ƒ'(x) = (tan x)'.
Verify the formula (x3)' = 3x2 by writing x3 = x · x · x and applying the Product Rule.
Show that the tangent lines at x = 1 ± √2 to the conchoid with equation (x − 1)2(x2 + y2) = 2x2 are vertical (Figure 11). 2. -1 -2 1 2 X
In x and y are functions of a variable t. Use implicit differentiation to express dy/dt in terms of dx/dt, x, and y.x2y = 3
Find all critical points of the function.g(θ) = sin2θ
Find all critical points of the function.R(θ) = cos θ + sin2θ
Let ƒ(x) = 2x2 − 8x + 7.(a) Find the critical point c of ƒ and compute ƒ(c).(b) Find the extreme values of ƒ on [0, 5].(c) Find the extreme values of ƒ on [−4, 1].
State whether ƒ(x) = x−1 (Figure 19) has a minimum or maximum value on the following intervals:(a) (0, 2) (b) (1, 2) (c) [1, 2] 1 2 +X 3
Find all critical points of the function.ƒ(x) = x2 − 2x + 4
Find all critical points of the function. f(x) = x³2x²-54x + 2
Find all critical points of the function.ƒ(x) = 7x − 2
Find all critical points of the function.ƒ(t) = 8t3 − t2
Find all critical points of the function. g(z) || 1 z-1 1 IN 2
Find all critical points of the function.ƒ(x) = x−1 − x−2
Find all critical points of the function. f(x) = x² + 1
Find all critical points of the function. 好 f(x) = x² - 4x + 8
Find all critical points of the function.ƒ(t) = t − 4 √t + 1
Find all critical points of the function.ƒ(t) = 4t − √t2 + 1
Find all critical points of the function.ƒ(x) = x2 √1 − x2
Find all critical points of the function.ƒ(x) = x + |2x + 1|
Find the extreme values of ƒ(x) = 2x3 − 9x2 + 12x on [0, 3] and [0, 2].
Compute the critical points of h(t) = (t2 − 1)1/3. Check that your answer is consistent with Figure 20. Then find the extreme values of h on [0, 1] and on [0, 2]. -2 - 1 h(t) 1- -1 1 2
Find the critical points of ƒ(x) = sin x + cos x and determine the extreme values on [0, π/2].
Plot ƒ(x) = 4 √x − 2x + 3 on [0, 3] and indicate where it appears that the minimum and maximum occur. Then determine the minimum and maximum using calculus.
Plot ƒ(x) = 2x3 − 9x2 + 12x on [0, 3] and locate the extreme values graphically. Then verify your answer using calculus.
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = 2x2 + 4x + 5, [−2, 2]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = 2x2 + 4x + 5, [0, 2]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = 6t − t2, [0, 5]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = 6t − t2, [4, 6]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = x3 − 6x2 + 8, [1, 6]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = x3 − 6x2 + 8, [−1, 6]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = x3 − 6x2 + 8, [1, 3]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = x3 − 6x2 + 8, [−1, 3]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = x3 − 12x2 + 21x, [0, 2]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = z5 − 80z, [−3, 3]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints. y = x² +1 x-4 [5,6]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = 2x5 + 5x2, [−2, 2]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints. y = 1- x x² + 3x² [1,4]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints. y = x- 4x x+1' [0, 3]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = 2 √x2 + 1 − x, [0, 2]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = (2 + x)√2 + (2 − x)2, [0, 2]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = √1 + x2 − 2x, [0, 1]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = √x + x2 − 2 √x, [0, 4]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = (t − t2)1/3, [−1, 2]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = sin x cos x, [0, π/2]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = x + sin x, [0, 2π]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = √2 θ − sec θ, [0, π/3]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.x4 − 2x2 + 1, [−3, 3]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = x3 + x2 − x, [−2, 2]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = cos θ + sin θ, [0, 2π]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = θ − 2 sinθ, [0, 2π]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = 4 sin3 θ − 3 cos2 θ, [0, 2π]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = tan x − 2x, [0, 1]
Find the minimum and maximum values of the function on the given interval by comparing values at the critical points and endpoints.y = sec2x − 2 tan x, [−π/6, π/3]
Plot ƒ(x) = 2 + x2/x on (0, 5) and use the graph to explain why there is a minimum value, but no maximum value, of ƒ on (0, 5). Use calculus to find the minimum value.
Plot ƒ(x) = 4x − 1 − x2 / x on (0, 3) and use the graph to explain why there is a maximum value, but no minimum value, of ƒ on (0, 3). Use calculus to find the maximum value.
Let ƒ(θ) = 2 sin 2θ + sin 4θ.(a) Show that θ is a critical point if cos 4θ = − cos 2θ.(b) Show, using a unit circle, that cos θ1 = − cos θ2 if and only if θ1 = π ± θ2 + 2πk for an integer k.(c) Show that cos 4θ = − cos 2θ if and only if θ = π/2+ πk or θ = π/6
Find the critical points and the extreme values on [0, 4]. In Exercises 59 and 60, refer to Figure 21.y = |x − 2| 30 20 10 -6 2 y = |x2 + 4x-12| -FIN П y = |cos x| П 2 П Зл 2 X
Find the critical points of ƒ(x) = 2 cos 3x + 3 cos 2x in [0, 2π]. Check your answer against a graph of ƒ.
Find the critical points and the extreme values on [0, 4]. In Exercises 59 and 60, refer to Figure 21.y = |3x − 9| 30 20 10 -6 2 y = |x2 + 4x-12| -FIN П y = |cos x| П 2 П Зл 2 X
Find the critical points and the extreme values on [0, 4]. In Exercises 59 and 60, refer to Figure 21.y = |x2 + 4x − 12| 30 20 10 -6 2 y = |x2 + 4x-12| -FIN П y = |cos x| П 2 П Зл 2 X $30 20 10 -6 2 y = |x2 + 4x-12| -FIN y = \cos x| 2 2 X$
Verify Rolle’s Theorem for the given interval by checking ƒ(a) = ƒ(b) and then finding a value c in (a, b) such that ƒ'(c) = 0. THEOREM 4 Rolle's Theorem Assume that f is continuous on [a, b] and differen- tiable on (a, b). If f(a) = f(b), then there exists a number c between a and b such that
Find the critical points and the extreme values on [0, 4]. In Exercises 59 and 60, refer to Figure 21.y = | cos x| 30 20 10 -6 2 y = |x2 + 4x-12| -FIN П y = \cos x| П 2 П Зл 2 X $30 20 10 -6 2 y = |x2 + 4x-12| -FIN y = \cos x| 2 2 X$
Verify Rolle’s Theorem for the given interval by checking ƒ(a) = ƒ(b) and then finding a value c in (a, b) such that ƒ'(c) = 0. THEOREM 4 Rolle's Theorem Assume that f is continuous on [a, b] and differen- tiable on (a, b). If f(a) = f(b), then there exists a number c between a and b such that
Verify Rolle’s Theorem for the given interval by checking ƒ(a) = ƒ(b) and then finding a value c in (a, b) such that ƒ'(c) = 0. THEOREM 4 Rolle's Theorem Assume that f is continuous on [a, b] and differen- tiable on (a, b). If f(a) = f(b), then there exists a number c between a and b such that
Verify Rolle’s Theorem for the given interval by checking ƒ(a) = ƒ(b) and then finding a value c in (a, b) such that ƒ'(c) = 0. THEOREM 4 Rolle's Theorem Assume that f is continuous on [a, b] and differen- tiable on (a, b). If f(a) = f(b), then there exists a number c between a and b such that
Prove that ƒ(x) = x4 + 5x3 + 4x has no root c satisfying c > 0. x = 0 is a root and apply Rolle’s Theorem. THEOREM 4 Rolle's Theorem Assume that f is continuous on [a, b] and differen- tiable on (a, b). If f(a) = f(b), then there exists a number c between a and b such that f'(c) = 0.
Prove that (x) = ƒx3 + 3x2 + 6x has precisely one real root.
In 1919, physicist Alfred Betz argued that the maximum efficiency of a wind turbine is around 59%. If wind enters a turbine with speed v1 and exits with speed v2, then the power extracted is the difference in kinetic energy per unit time:where m is the mass of wind flowing through the rotor per $V1 V2 (A) Wind flowing through a turbine. 0.6 0.5 F 0.4+ 0.3 0.2+ 0.1 0.5 (B) F is the fraction of energy$
The concentration C(t) (in milligrams per cubic centimeter) of a drug in a patient’s bloodstream after t hours isFind the maximum concentration in the time interval [0, 8] and the time at which it occurs. C(t) = 0.016t 1² + 4t + 4
Prove that x = 4 is the greatest root of ƒ(x) = x4 − 8x2 − 128.
The Bohr radius a0 of the hydrogen atom is the value of r that minimizes the energy 2 1- E(r) = where ħ, m, e, and o are physical constants. Show that ao = 4лεħ²/(me²). Assume that the minimum occurs at a critical point, as suggested by Figure 23. E(r) (10-18 joules) -
The response of a circuit or other oscillatory system to an input of frequency ω (“omega”) is described by the function 50 φ(ω) = @₂ (A) D = 0.01 Both wo (the natural frequency of the system) and D (the damping factor) are positive constants. The graph of ø is called a resonance curve,
Find the maximum of y = x − xn on [0, 1], where n > 1.
Sketch the graph of a continuous function on (0, 4) with a minimum value but no maximum value.
Sketch the graph of a function on [0, 4] having(a) Two local maxima and one local minimum(b) An absolute minimum that occurs at an endpoint, and an absolute maximum that occurs at a critical point
A rainbow is produced by light rays that enter a raindrop (assumed spherical) and exit after being reflected internally as in Figure 25. The angle between the incoming and reflected rays is θ = 4r − 2i, where the angle of incidence i and refraction r are related by Snell’s Law sin i = n sin r
Sketch the graph of a function ƒ on [0, 4] with a discontinuity such that f has an absolute minimum but no absolute maximum.
Generalize Exercise 87: Show that if the horizontal line y = c intersects the graph of ƒ(x) = x2 + rx + s at two pointsFigure 26Data From Exercise 87Show that if the quadratic polynomial ƒ(x) = x2 + rx + s takes on both positive and negative values, then its minimum value occurs at the midpoint
Show, by considering its minimum, that ƒ(x) = x2 − 2x + 3 takes on only positive values. More generally, find the conditions on r and s under which the quadratic function ƒ(x) = x2 + rx + s takes on only positive values. Give examples of r and s for which f takes on both positive and negative

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