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mathematics
calculus early transcendentals 9th

Questions and Answers of Calculus Early Transcendentals 9th

Test the series for convergence or divergence. √j j + 5 Σ (-1) - j=1
Determine whether the series is convergent or divergent. If it is convergent, find its sum. 00 Σ n=1 1 4ten
Determine whether the sequence converges or diverges. If it converges, find the limit.an = n1/n
Determine whether the sequence converges or diverges. If it converges, find the limit.an = 2−n cos nπ
Determine whether the sequence converges or diverges. If it converges, find the limit. ann-√√n+ 1 √n +3
Suppose thatis known to be a convergent series. Prove thatis a divergent series. Σ=1 in=1 n (an # (0 =
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.For any vectors u and v in V3, |u × v| = |v
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.The vector (3, −1, 2) is parallel to the
Find parametric equations for the line.The line through (−2, 2, 4) and perpendicular to the plane 2x − y + 5z = 12
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If u × v = 0, then u = 0 or v = 0.
Find the angle between the vectors.a = 8i − j + 4k, b = 4j + 2k
Use traces to sketch and identify the surface.y = z2 − x2
Find the length of the curve.r(t) = i + t2j + t3k, 0 ≤ t ≤ 1
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If |r(t)| = 1 for all t, then r'(t) is
Evaluate the integral. [i+te²j + √ [ k) + √ik) di dt 1 1 + 1²
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.The osculating circle of a curve C at a point
Draw the projection of the curve onto the given plane.r(t) = (t + 1, 3t + 1, cos(t/2)), xy-plane
Use a computer algebra system to compute the curvature function std. Then graph the space curve and its curvature function. Comment on how the curvature reflects the shape of the curve. r(t) = (te',
The graph of a function f is shown. Find an equation of the tangent plane to the surface z = f(x, y) at the specified point.f(x, y) = 16 − x2 − y2 XY ZA (2, 2, 8) z = 16-x² - y² y
Suppose that lim(x, y)→(3, 1) f(x, y) = 6. What can you say about the value of f(3, 1)? What if f is continuous?
Use Equation 5 to find dy/dx.cos(xy) = 1 + sin y 5 dy dx || aF ax ƏF ду Fx Fy
Find the linear approximation of the functionat the point (2, 3, 4) and use it to estimate the number f(x, y, z) = x³√√y² + z²
Find the differential of the function. = √x² + 3y² u =
Find the differential of the function.H = x2y4 + y3z5
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts (a) –
Find the critical numbers of the function.p(t) = te4t
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x ln 1 x-18 1 X
You may want to check your work with a graphing calculator or computer.f(x) = x3 – 3x2 + 4(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the
Find the critical numbers of the function.f(θ) = 2 cos θ + sin2θ
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim sin 5x csc 3x x→0
If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 32 inches, what diameter pizza will reward you with the largest slice?
Use the guidelines of this section to sketch the curve.y = e–x sin x, 0 ≤ x ≤ 2π
Find the critical numbers of the function.f(θ) = θ + √2 cos θ
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim √√xe-x/2 x- ▶00
Find the critical numbers of the function.f(x) = x1/3(4 − x)2/3
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x sin(π/x)
The graph of a function y = f(x) is shown. At which point(s) are the following true?a.b.c. dy d'y and are both positive. dx dx²
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. x - sin x lim x→0 x
Find the critical numbers of the function.h(x) = x-1/3(x − 2)
Use the graphs of f, f', and f" to estimate the x-coordinates of the maximum and minimum points and inflection points of f.f(x) = e–0.1x ln(x2 – 1)
Find the critical numbers of the function.F(x) = x4/5(x − 4)2
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim 0-x cos x 1 + x² x4
Use the graphs of f, f', and f" to estimate the x-coordinates of the maximum and minimum points and inflection points of f. f(x) = cos²x x² + x + 1 2 -T≤x≤ T
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x →∞0 e-x (π/2) -
(a) Graph the function f(x) = 1/(1 + e1/x).(b) Explain the shape of the graph by computing the limits of f(x) as x approaches ∞, –∞, 0+, and 0–.(c) Use the graph of f to estimate the
Find the critical numbers of the function.h(t) = t 3/4 − 2t1/4
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. - x − 1 1' lim x1 x , b 0
Use the guidelines of this section to sketch the curve.y = csc x – 2sin x, 0 < x < π
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. x² sin x lim x0 sin x x
The figure shows graphs (in blue) of several members of the family of polynomials f(x) = cx4 – 4x2 + 1.(a) For which values of c does the curve have minimum points?(b) Show that the minimum and
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x→0+ arctan 2x In x
Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f' and f" to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and
Let f(x) = 1/x andShow that f'(x) = g'(x) for all x in their domains. Can we conclude from Corollary 7 that f – g is constant? g(x) X 1 + 1 -| X if x > 0 if x < 0
Use the guidelines of this section to sketch the curve.y = 2x – tan x, –π/2 < x < π/2
A rectangle has its base on the x-axis and its upper two vertices on the = parabola y = 4 – x2. What is the largest possible area of the rectangle?
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. x sin(x - 1) lim x12x² -
Use the guidelines of this section to sketch the curve.y = x tan x, –π/2 < x < π/2
Sketch the graph of a function that satisfies all of the given conditions.(a) f'(x) > 0 and f"(x) < 0 for all x(b) f'(x) < 0 and f"(x) > 0 for all x
If one side of a triangle has length a and another has length 2a, show that the largest possible area of the triangle is a2.
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. In(1 + x) lim x-0 cos x +
Use the guidelines of this section to sketch the curve.y = x + cos x
Sketch the graph of a function that satisfies all of the given conditions.(a) f'(x) < 0 and f"(x) < 0 for all x(b) f'(x) > 0 and f"(x) > 0 for all x
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x-0 et + et 2 cos
Find the critical numbers of the function.A(x) = |3 − 2x|
Use the guidelines of this section to sketch the curve.y = sin3x
Suppose f" is continuous on (– ∞, ∞).(a) If f'(2) = 0 and f"(2) = –5, what can you say about f ?(b) If f'(6) = 0 and f"(6) = 0, what can you say about f ?
Find the critical numbers of the function.g(t) = t5 + 5t3 + 50t
Find the critical numbers of the function.f(x) = 2x3 + x2 + 8x
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim 818 (In x)² X
Suppose that f and t are continuous on [a, b] and differentiable on (a, b). Suppose also that f(a) = g(a) and f'(x), g'(x) for a < x < b. Prove that f(b) < g(b).
Find the critical numbers of the function.f(x) = 3x4 + 8x3 − 48x2
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x->0 sin ¹x X
Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.
Use the guidelines of this section to sketch the curve.y = x5/3 – 5x2/3
Find the critical numbers of the function.g(v) = v3 − 12v + 4
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. x - sin x tan x lim x→0 x
Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer?f(x) = 1 + 3x2 – 2x3
Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x→0 tanh x tan x
Use the guidelines of Section 4.5 to sketch the curve.y = ex sin x, –π ≤ x ≤ π
Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by looking at a graph to find initial approximations. In(x2 + 2) = 3x Vx² + 1
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x-0 sinh x − x - x3
Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by looking at a graph to find initial approximations. √√4x³ = ex²
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x-o et + e* - 2 ex- x
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim e"/10 3 И
Show that the equation x4 + 4x + c = 0 has at most two real solutions.
(a) Graph the function.(b) Explain the shape of the graph by computing the limit as x → 0+ or as x → ∞.(c) Estimate the maximum and minimum values and then use calculus to find the exact
Use the guidelines of Section 4.5 to sketch the curve. y = √√1 -x + √1 + x
Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 2x² + 5 x² + 1
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim x-0 √1 + 2x -√I- 1
Show that the equation x3 – 15x + c = 0 has at most one solution in the interval [–2, 2].
Use the guidelines of Section 4.5 to sketch the curve. y (x - 1)³ x² 2
Find the most general antiderivative of the function. (Check your answer by differentiation.)f(x) = 2x + 4 sinh x
Refer to Exercise 23. Find the dimensions of the cylindrical mailing tube of greatest volume that may be mailed using the US postal service.Data From Exercise 23:A package to be mailed using the US
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. lim 1-0 8' - 5' t
Show that the equation has exactly one real solution. x3 + ex = 0
Use the guidelines of Section 4.5 to sketch the curve. y 1 x ² 1 (x - 2)²
Find the most general antiderivative of the function. (Check your answer by differentiation.) g(v) = 2 cos v 3 √1 - v²
A package to be mailed using the US postal service may not measure more than 108 inches in length plus girth. (Length is the longest dimension and girth is the largest distance around the package,
Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. In(x/3) lim x3 3x

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