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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

From the vertex (0, c) of the catenary y = c cosh(x/c) a line L is drawn perpendicular to the tangent to the catenary at point P. Prove that the length of L intercepted by the axes is equal to the
Show thatarctan(sinh x) = arcsin(tanh x).
In Exercises prove the differentiation formula. d. [cosh x] = sinh x dx
Let x > 0 and b > 0. Show that b J-b et dt 2 sinh bx X
In Exercises prove the differentiation formula. d dx [coth x] = -csch² x
In Exercises verify the differentiation formula. 1 -[sinh~1x]= √x² + 1 2 d dx
In Exercises prove the differentiation formula. d [sech x] = - sech x tanh x dx
In Exercises verify the differentiation formula. d dx -[sech-¹x] = = 1 x√1-x²
In Exercises find the indefinite integral. 3 2√x(1+x) -dx
(a) Write an integral that represents the area of the region in the figure.(b) Use the Trapezoidal Rule with n = 8 to estimate the area of the region.(c) Explain how you can use the results of parts
In Exercises verify the rule by differentiating. Let a > 0. S₁² du a² + u² 1 a n arctan + C a
Graphon [0, 10]. Prove that Y₁ || X 25 1 + x²⁹ y2 = arctan x, and y3 = x
In Exercises verify the rule by differentiating. Let a > 0. S du a² - u² u arcsin + C a
In Exercises verify the rule by differentiating. Let a > 0. Sur И du u√√√u² - a² - a |n| arcsec + C a
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.One way to findis to use the Arcsine Rule. 2e2x 9-e2r dx 2x
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. dx √4x² - arccos + C
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. dx 25 + x² || 1 25 arctan X 25 + C
In Exercises find the indefinite integral. x + 5 9- (x - 3)² dx
In Exercises find the indefinite integral. t 1 + 25 dt
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. dx 3x√√9x² - 16 1 4 3x arcsec + C 4
Consider the integral(a) Find the integral by completing the square of the radicand.(b) Find the integral by making the substitution u = √x.(c) The antiderivatives in parts (a) and (b) appear to
In Exercises find the indefinite integral. L dx /9 - x²
(a) Sketch the region whose area is represented by(b) Use the integration capabilities of a graphing utility to approximate the area.(c) Find the exact area analytically. So arcsin x dx.
(a) Show that(b) Approximate the number π using Simpson's Rule (with n = 6) and the integral in part (a).(c) Approximate the number π by using the integration capabilities of a graphing utility.
In Exercises find the area of the region. || 2 x² + 4x + 8 0.5 0.2- 0.1 -5 -4 -3 -2 -1 X
In Exercises find the area of the region. 1 y x² - 2x + 5 -2 0.4 0.3 0.2 -0.2 y 2 3 4 نرا X
In Exercises find the area of the region. -2 4ex 1 + e2x 1 3 y x = ln √3 2 X
In Exercises find the area of the region. y 3 cos x 1 + sin² x y 元|4 3, - -2 -3+ RI4 π T 2 X
In Exercises find the area of the region. y = -2 2 √4x² 1 3 2 -1 y 1 2
In Exercises use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition. dy dx y(0) = 2 2y √16x² ||
In Exercises find the area of the region. y 2 y 1 x√√√x² X = - 1 2 2 X
In Exercises use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition. dy dx 10 x√√√x² - 1 y (3) =
In Exercises a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the
In Exercises a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the
In Exercises use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition. dy dx y(4) = 2 1 12 + x² ||
In Exercises use the differential equation and the specified initial condition to find y. dy dx y(2) = - 1 4 + x² ||
In Exercises use the differential equation and the specified initial condition to find y. dy dx || 1 √4 - x² y(0) = -
In Exercises use the specified substitution to find or evaluate the integral. Sz u dx 2√3x√√√x + 1 √x + 1
In Exercises determine which of the integrals can be found using the basic integration formulas you have studied so far in the text.(a)(b)(c) S x - 1 dx
Using the graph, which value best approximates the area of the region between the x-axis and the function over the interval [-1/2, 1/2]? Explain.(a) -3 (b) 1/2(c) 1(d) 2(e) 4
Decide whether you can find the integralusing the formulas and techniques you have studied so far. Explain your reasoning. 2 dx x² + 4
In Exercises determine which of the integrals can be found using the basic integration formulas you have studied so far in the text.(a)(b)(c) ex² dx
In Exercises determine which of the integrals can be found using the basic integration formulas you have studied so far in the text.(a)(b)(c) 1 1 + x4 dx
In Exercises use the specified substitution to find or evaluate the integral. 3 dx √x(1 + x) u = √√√x
In Exercises determine which of the integrals can be found using the basic integration formulas you have studied so far in the text.(a)(b)(c) 1 1 - x2 dx
In Exercises find or evaluate the integral by completing the square. X 9 + 8x² - =dx x4
In Exercises find or evaluate the integral by completing the square. X x4 + 2x² + 2 dx
In Exercises use the specified substitution to find or evaluate the integral. √√x-20 x + 1 S= U = - dx x-2
In Exercises use the specified substitution to find or evaluate the integral. Ive- u = et - 3 dt √et - 3
In Exercises find or evaluate the integral by completing the square. S 2 2x - 3 /4x = x² dx
In Exercises find or evaluate the integral by completing the square. 2 -x² + 4x dx
In Exercises find or evaluate the integral by completing the square. Sa- 1 (x - 1)√√x² - 2x = dx
In Exercises find or evaluate the integral by completing the square. 1 -x² - 4x 2 dx
In Exercises find or evaluate the integral by completing the square. 2x - 5 x² + 2x + 2 dx
In Exercises find or evaluate the integral by completing the square. JO dx x² - 2x + 2
In Exercises evaluate the definite integral. 1/√2 arccos x /1-x² dx
In Exercises find or evaluate the integral by completing the square. 2x x² + 6x + 13 dx
In Exercises find or evaluate the integral by completing the square. 1-2 dx x² + 4x + 13
In Exercises evaluate the definite integral. (π/2 COS X 1 + sin² x dx
In Exercises evaluate the definite integral. TT Jπ/2 sin x 1 + cos²x dx
In Exercises evaluate the definite integral. 1/√√/2 arcsin x √1-x² dx
In Exercises evaluate the definite integral. In 4 JIn 2 e-x 1-e-2r dx
In Exercises evaluate the definite integral. In 5 ex 1 + e2x dx
In Exercises evaluate the definite integral. S₁ = 1 16x² - 5 dx
In Exercises evaluate the definite integral. √√3/2 0 1 1 + 4x² dx
In Exercises evaluate the definite integral. *6 J3 1 25+ (x - 3)² dx
In Exercises evaluate the definite integral. √2 Jo 1 √4x² dx
In Exercises evaluate the definite integral. 3 1 √√3x√√4x² - 9 dx
In Exercises evaluate the definite integral. (1/6 J0 3 19x² dx
In Exercises find the indefinite integral. x - 2 (x + 1)² + 4' 7 dx
In Exercises find the indefinite integral. x² + 3 x√√x² - 4 X- dx
In Exercises find the indefinite integral. sec² x 25- tan²x =dx
In Exercises find the indefinite integral. x-3 x² + 1 dx
In Exercises find the indefinite integral. 1 √x√1-x dx
In Exercises find the indefinite integral. sin x 7 + cos²x dx
In Exercises find the indefinite integral. 2 x √9x² - 25 dx
In Exercises find the indefinite integral. 4 + 4x dx
In Exercises find the indefinite integral. SIVT 1 x√1 - (In x)² Jz dx
In Exercises find the indefinite integral. 1 4 + (x − 3)2 dx
In Exercises find the indefinite integral. 1 x√x - 4 =dx
In Exercises find the indefinite integral. I 1 t dt
In Exercises find the indefinite integral. 1 1- (x + 1)² dx
In Exercises use a computer algebra system to graph the slope field for the differential equation and graph the solution satisfying the specified initial condition. dy dx y(0) = 4 || 心 1 + x²
In Exercises find the indefinite integral. 12 1 + 9x² dx
In Exercises find the indefinite integral. 1 √√4x² /4x² - 1 dx
In Exercises find the indefinite integral. dx /1-4x²
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. arcsin 프 4 √/2 2
Prove that arcsin x = arctan X 1-x² |x < 1.
In the figure, find PR such that 0 ≤ PR ≤ 3 and m ∠ θ is a maximum. 3 Re 2 5
Some calculus textbooks define the inverse secant function using the range [0, π/2) U [π, 3π/2).(a) Sketch the graph y = arcsec x using this range.(b) Show that y' 1 -2 x√x² - 1 -
Prove each differentiation formula.(a)(b)(c)(d) d dx -[arctan u] u 1 + u²
In the figure, find the value of c in the interval [0, 4] on the x-axis that maximizes angle θ. y (0, 2) 0 C (4,2)
(a) Prove that arctan x + arctan y = arctan(b) Use the formula in part (a) to show that x + y 1- xy' xy # 1.
A billboard 85 feet wide is perpendicular to a straight road and is 40 feet from the road (see figure). Find the point on the road at which the angle θ subtended by the billboard is a maximum.
In a free-fall experiment, an object is dropped from a height of 256 feet. A camera on the ground 500 feet from the point of impact records the fall of the object (see figure).(a) Find the position
A patrol car is parked 50 feet from a long warehouse (see figure). The revolving light on top of the car turns at a rate of 30 revolutions per minute. Write θ as a function of x. How fast is the
Repeat Exercise 91 for an altitude of 3 miles and describe how the altitude affects the rate of change of θ.Data from in Exercise 91An airplane flies at an altitude of 5 miles toward a point
It Use a graphing utility to graph ƒ(x) = sin x and g(x) = arcsin(sin x).(a) Why isn't the graph of g the line y = x?(b) Determine the extrema of g.
An airplane flies at an altitude of 5 miles toward a point directly over an observer. Consider θ and x as shown in the figure.(a) Write θ as a function of x.(b) The speed of the plane is 400
A television camera at ground level is filming the lift-off of a rocket at a point 800 meters from the launch pad. Let be the angle of elevation of the rocket and let s be the distance between the

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