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mathematics
calculus early transcendentals 9th
Calculus Early Transcendentals 9th Edition James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin - Solutions
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.x2 – y2 + 2y = 2
The position of an object in circular motion is modeled by the given parametric equations, where t is measured in seconds. How long does it take to complete one revolution? Is the motion clockwise or counterclockwise?x = 5 cos t, y = −5 sin t
Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter.x = t3 + 6t + 1, y = 2t – t2; t = –1
Find an equation for the hyperbola. Then find the foci and asymptotes. у -1 0 1 x
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.9y2 – 4x2 – 36y – 8x = 4
Graph the conicswith e = 0.4, 0.6, 0.8, and 1.0 on a common screen. How does the value of e affect the shape of the curve? e 1 - e cos 0
Find an equation for the hyperbola. Then find the foci and asymptotes. ГУА 4 0 4 x
Find dy/dx and d2y/dx2.x = t + sin t, y = t – cos t
Identify the type of conic section whose equation is given and find the vertices and foci.4x2 = y2 + 4
Find dy/dx and d2y/dx2.x = 1 + t2, y = t – t3
Identify the type of conic section whose equation is given and find the vertices and foci.4x2 = y + 4
Identify the type of conic section whose equation is given and find the vertices and foci.x2 = 4y – 2y2
Find the area of the region that lies inside both curves.r = 3 sin θ, r = 3 cos θ
Find the area enclosed by the loop of the curve in Exercise 29.Data From Exercise 29:Use a graph to estimate the coordinates of the lowest point on the curve x = t3 – 3t, y = t2 + t + 1. Then use calculus to find the exact coordinates.
Identify the type of conic section whose equation is given and find the vertices and foci.y2 – 2 = x2 – 2x
The figure shows a graph of r as a function of θ in Cartesian coordinates. Use it to sketch the corresponding polar curve. "" 0 -1 -2 ´ 2n ម
Identify the type of conic section whose equation is given and find the vertices and foci.3x2 – 6x – 2y = 1
Find the area enclosed by the curve in Exercise 31.Data From Exercise 31:At what points does the curvex = 2a cos t – a cos 2t y = 2a sin t – a sin 2thave vertical or horizontal tangents? Use this information to help sketch the curve.
Identify the type of conic section whose equation is given and find the vertices and foci.x2 – 2x + 2y2 – 8y + 7 = 0
Find the area enclosed by the given parametric curve and the x-axis.x = t3 + 1, y = 2t − t2 у 4 0 X
Jupiter’s orbit has eccentricity 0.048 and the length of the major axis is 1.56 × 109 km. Find a polar equation for the orbit.
Use a graphing calculator or computer and the result of Exercise 37(a) to draw the triangle with vertices A(1, 1), B(4, 2), and C(1, 5).Data From Exercise 37:(a) Show that the parametric equationsx = x1 + (x2 − x1)t y = y1 + (y2 − y1)twhere 0 ≤ t ≤ 1, describe the line segment
Find an equation for the conic that satisfies the given conditions.Parabola, vertical axis, passing through (0, 4), (1, 3), and (–2, –6)
Find the length of the curve.x = 3t2, y = 2t3, 0 ≤ t ≤ 2
Find all points of intersection of the given curves.r = 2 sin 2θ, r = 1
Find parametric equations for the position of a particle moving along a circle as described.The particle travels counterclockwise around a circle with center (1, 3) and radius 1 and completes a revolution in three seconds.
Find the length of the curve.x = 2 + 3t, y = cosh 3t, 0 ≤ t ≤ 1
Find the length of the curve.r = 1/θ, π ≤ θ ≤ 2π
Find the length of the curve.r = sin3(θ/3), 0 ≤ θ ≤ π
The position (in meters) of a particle at time t seconds is given by the parametric equations(a) Find the speed of the particle at the point (6, –4).(b) What is the average speed of the particle for 0 ≤ t ≤ 8 ? x = 1/(t² + 3) y = 5-t³
Find an equation for the conic that satisfies the given conditions.Ellipse, foci (0, –1), (8, –1), vertex (9, –1)
(a) Find the exact length of the portion of the curve shown in blue.(b) Find the area of the shaded region. 0 r = 2 cos²(0/2)
Find an equation for the conic that satisfies the given conditions.Ellipse, center (–1, 4), vertex (–1, 0), focus (–1, 6)
Find an equation for the conic that satisfies the given conditions.Ellipse, foci (±4, 0), passing through (–4, 1.8)
Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices (±3, 0), foci (±5, 0)
Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices (–1, 2), (7, 2), foci (–2, 2), (8, 2)
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.r = sin (θ/2)
Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices (±3, 0), asymptotes y = ±2x
Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.r = cos (θ/3)
Find an equation for the conic that satisfies the given conditions.Hyperbola, foci (2, 0), (2, 8), asymptotes y = 3 + 1/2 x and y = 5 – 1/2 x
Suppose that the position of each of two particles is given by parametric equations. A collision point is a point where the particles are at the same place at the same time. If the particles pass through the same point but at different times, then the paths intersect but the particles don’t
Graph the polar curve. Choose a parameter interval that produces the entire curve.r = 1 + 2 sin(θ/2) (nephroid of Freeth)
Find the area of the region enclosed by the hyperbola x2/a2 – y2/b2 = 1 and the vertical line through a focus.
The graph shows two red circles with centers (–1, 0) and (1, 0) and radii 3 and 5, respectively. Consider the collection of all circles tangent to both of these circles. (Some of these are shown in blue.) Show that the centers of all such circles lie on an ellipse with foci (±1, 0). Find an
Here we investigate the reflection properties of ellipses and hyperbolas.Let P(x1, y1) be a point on the ellipse x2/a2 + y2/b2 = 1 with foci F1 and F2 and let α and β be the angles between the lines PF1, PF2 and the ellipse as shown in the figure. Prove that α = β. This explains how whispering
Here we investigate the reflection properties of ellipses and hyperbolas.Let P(x1, y1) be a point on the hyperbola x2/a2 – y2/b2 = 1 with foci F1 and F2 and let α and β be the angles between the lines PF1, PF2 and the hyperbola as shown in the figure. Prove that α = β. This shows that
Evaluate the integral, if it exists. =dx 1- x4
Use the properties of integrals to verify the inequality. =/2 sin x T/4 x dx VI 2
Findif So f(x) dx
Evaluate the definite integral. 50 (3t - 1)5⁰ dt
Evaluate the definite integral. Jo 3 /1 + 7x dx
Use the properties of integrals to verify the inequality. fx sin ¹x dx = π/4
Evaluate the definite integral. 273³csc²(t) dt JT/3
If where f is the function whose graph is given, which of the following values is largest?(A) F(0)(B) F(1)(C) F(2)(D) F(3)(E) F(4) F(x) = f(t) dt,
Evaluate the definite integral. =/6 sin t cos²t Jo dt
What is wrong with the equation? L₂x+dx==1₁ -3 xlw 8 -2
Evaluate the definite integral. √√2 + √√x √x - dx
What is wrong with the equation? 2 4 S²₁ -dx = -1 X 2 2 ³/1- X² 3 2
Evaluate the definite integral. 3 el/x xp - x² 2
Evaluate the definite integral. et S Jo 1 + e²x -dx
Evaluate the definite integral. *π/4 557144 (x³ + x¹ tan x) dx -π/4
If h(t) is a person’s heart rate in beats per minute t minutes into an exercise session, what does ∫300 h(t) dt represent?
Evaluate the definite integral. TT/2 0 cos x sin(sin x) dx
If f is continuous and ∫20 f(x) dx = 6, evaluate f/2 f(2 sin 0) cos 0 de Jo
Evaluate the definite integral. dx 70 √(1 + 2x)² EIJ
Use Property 8 of integrals to estimate the value of the integral. 1 S'₁ x x³ dx Jo
The Fresnel function S(x) = ∫x0 sin(1/2πt2) dt was introduced in Section 5.3. Fresnel also used the functionin his theory of the diffraction of light waves.(a) On what intervals is C increasing?(b) On what intervals is C concave upward?(c) Use a graph to solve the following equation correct to
Use Property 8 of integrals to estimate the value of the integral. 0 1 dx x +41
The velocity function (in m/s) is given for a particle moving along a line. Find(a) The displacement(b) The distance traveled by the particle during the given time interval.ν(t) = 3t – 5, 0 ≤ t ≤ 3
Use Property 8 of integrals to estimate the value of the integral. TT/3 Jπ/4 tan x dx
Evaluate the definite integral. √x √x² + a² dx (a > 0)
The velocity function (in m/s) is given for a particle moving along a line. Find(a) The displacement(b) The distance traveled by the particle during the given time interval.ν(t) = t2 – 2t – 3, 2 ≤ t ≤ 4
Evaluate the definite integral. Cπ/3 J-#/3 x¹ sin x dx
Use Property 8 of integrals to estimate the value of the integral. ² (x³ - 3x + 3) dx Jo
If f(x) = ∫x0 (1 – t2)et2 dt, on what interval is f increasing?
If f is a continuous function such thatfor all x, find an explicit formula for f(x). f* ƒ (1) dt = (x − 1)e²¹ + $* e¯¹ƒ (1) dt 1
Evaluate the definite integral. 2 f²x√x – 1 dx -
Use Property 8 of integrals to estimate the value of the integral. 2 fxe xex dx
Evaluate the definite integral. Jo x √1 + 2x dx
Use Property 8 of integrals to estimate the value of the integral. 2T 2* (x - 2 sin x) dx TT
If f' is continuous on [a, b], show that 2 f f(x) f'(x) dx = [ƒ(b)]² − [ƒ(a)]²
Let F(x) = ∫x2et2 dt. Find an equation of the tangent line to the curve y = F(x) at the point with x-coordinate 2.
Evaluate the definite integral. dx x√ ln x Set-
If find g"(π/6). f(x) = finx √1 + 1² dt and g(y) = f' f(x) dx,
Use 1’Hospital’s Rule to evaluate the limit. 1/2 S ² lim x-0 x 2t 3 √₁³ + 1 dt
Evaluate the definite integral. S²(x - 1)e(x-1)² dx
If f is continuous on [0, 1], prove that f' f(x) dx = f' f(1-x) dx Jo
Evaluate the definite integral. e² + 1 Jo e² + z dz
Use 1’Hospital’s Rule to evaluate the limit. 1 lim x→∞ X Jo In(1 + e') dt
Evaluate the definite integral. 1 (x + 1)√x S₁ = dx
Evaluate the definite integral. dx JO (1 + / +
The graph of a car’s acceleration a(t), measured in ft/s2, is shown. Use the Midpoint Rule to estimate the increase in the velocity of the car during the six-second time interval. an 12 8 4 0 2 4 6 t (seconds)
The error functionis used in probability, statistics, and engineering.(a) Show that(b) Show that the function y = ex2 erf(x) satisfies the differential equation y' = 2xy + 2/√π. 2 erf(x) 름ienda -12 2 dt ㅠ IL
Evaluate the definite integral. 16 x¹/2 1 + x3/4 dx
(a) For the function f shown in the graph, verify graphically that the following inequality holds:(b) Prove that the inequality from part (a) holds for any function f that is continuous on [a, b].(c) Show that |f" f(x) dx | ≤ f | f(x) | dx
The Fresnel Function The Fresnel function S was defined in Example 3 and graphed in Figures 7 and 8.(a) At what values of x does this function have local maximum values?(b) On what intervals is the function concave upward?(c) Use a graph to solve the following equation correct to two decimal
Let f(0) = 0 and f(x) = 1/x if 0 < x ≤ 1. Show that f is not integrable on [0, 1].
Shown is a graph of the electric power consumption in the New England states (Connecticut, Maine, Massachusetts, NewHampshire, Rhode Island, and Vermont) for October 22, 2010 (P is measured in gigawatts and t is measured in hours starting at midnight). Use the fact that power is the rate of change
If f is continuous and t and h are differentiable functions, show that d - Sh(s) f(t) dt = f (h(x)) h'(x) — ƒ (g(x)) g'(x) dx Jg(x)
(a) Show that(b) Show that 1 ≤ √1 + x³ ≤ 1 + x³ for x ≥ 0.
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