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mathematics
calculus early transcendentals 9th

Questions and Answers of Calculus Early Transcendentals 9th

Test the series for convergence or divergence. 00 n! enz [="
Determine whether the sequence converges or diverges. If it converges, find the limit.an = e2n/(n+2)
Determine whether the sequence converges or diverges. If it converges, find the limit. an 1 + 2 n
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. fy(a, b) = lim y-b f(a, y) - f(a, b) y-b
Find and sketch the domain of the function.f(x, y) = ln(x + y + 1)
Suppose a block of mass m is placed on an inclined plane, as shown in the figure. The block’s descent down the plane is slowed by friction; if θ is not too large, friction will prevent the block
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) = y2 sin x ZA (π/2, -2, 4) z = y² sin x y
Find and sketch the domain of the function. f(x, y)=√4x² - y² + √1-x²
If f(x, y) = x2y/(2x − y2), find(a) f(1, 3)(b) f(−2, −1)(c) f(x + h, y)(d) f(x, x)
Find dz/dt in two ways: by using the Chain Rule, and by first substituting the expressions for x and y to write z as a function of t. Do your answers agree?z = x2y + xy2, x = 3t,
If g(x, y) = x sin y + y sin x, find(a) g(π, 0)(b) g(π/2, π/4)(c) g(0, y)(d) g(x, y + h)
For what values of the number r is the functioncontinuous on R3? f(x, y, z) = (x + y + z)' x² + y² + z² 0 if (x, y, z) (0, 0, 0) if (x, y, z) = (0, 0, 0)
The temperature T (in °C) at a location in the Northern Hemisphere depends on the longitude x, latitude y, and time t, so we can write T = f (x, y, t). Let’s measure time in hours from the
Sketch the graph of the function.f(x, y) = 1 − y2
Let g(x, y) = x2 ln(x + y).(a) Evaluate g(3, 1).(b) Find and sketch the domain of t.(c) Find the range of g.
Use the Chain Rule to find dz/dt or dw/dt.z = xy3 − x2y, x = t2 + 1, y = t2 − 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.Dk f(x, y, z) = fz(x, y, z)
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 (2, 1) is a critical point of f andfxx(2,
Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 4.7.Exercise 3Find two positive numbers whose product is 100 and whose sum is a minimum.
Use Equation 5 to find dy/dx.y cos x = x2 + y2 5 dy dx || aF ax ƏF ду Fx Fy
Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 4.7.Exercise 8Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as
Find the local maximum and minimum values and saddle point(s) of the function. You are encouraged to use a calculator or computer to graph the function with a domain and viewpoint that reveals all
Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 4.7.Exercise 7Find the dimensions of a rectangle with perimeter 100 m whose area is as large as
Use the Squeeze Theorem to find the limit. lim (x, y)→→(0, 0) xy √x² + y²
Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 4.7.Exercise 18A box with a square base and open top must have a volume of 32,000 cm3. Find the
Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 4.7.Exercise 25Find the point on the line y = 2x + 3 that is closest to the origin.
Match the function with its graph (labeled I –VI). Give reasons for your choices.a.b.c.f(x, y) = ln(x2 + y2)d.e.f(x, y) = |xy|f.f(x, y) = cos(xyd f(x, y) = 1 1 + x² + y² 2
Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.Exercise 46Find the points on the surface y2 = 9 + xz that are closest to the origin.
Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.Exercise 47Find three positive numbers whose sum is 100 and whose product is a maximum.
Use Lagrange multipliers to give an alternate solution to the indicated exercise in Section 14.7.Exercise 54The base of an aquarium with given volume V is made of slate and the sides are made of
Find the radius of convergence and interval of convergence of the power series. - 21 - 0
Use the guidelines of this section to sketch the curve.y = ex/x2
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+ X 1 ex et - 1
(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) –
(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) –
Use the guidelines of Section 4.5 to sketch the curve. y=x√√2 + x
For the function f of Exercise 14, use a computer algebra system to find f' and f" and use their graphs to estimate the intervals of increase and decrease and concavity of f .Data From Exercise
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. x3x lim x-0 3 - 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. 3 lim x³e-x² x- ·00
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 x³/² sin(1/x) x
(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.g(x) = x2 ln x
Use the guidelines of this section to sketch the curve.y = (1 + ex)–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 In x tan(x/2) x-1+
Find the critical numbers of the function.B(u) = 4 tan-1u − u
(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) –
(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 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 cos x sec
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 x-1 1 In x
A formula for the derivative of a function f is given. How many critical numbers does f have?f'(x) = 5e−0.1| x | sinx − 1
Use the guidelines of this section to sketch the curve.y = ln(sin x)
(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) –
Use the guidelines of this section to sketch the curve.y = ln(1 + x3)
(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 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 (csc x x-0 cot x)
Find the absolute maximum and absolute minimum values of f on the given interval.f(x) = 12 + 4x − x2, [0, 5]
Use the guidelines of this section to sketch the curve.y = xe–1/x
(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
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 x¹/x 00 -X
Find the absolute maximum and absolute minimum values of f on the given interval.f(θ) = 1 + cos2θ, [π/4, π]
(a) If C(x) is the cost of producing x units of a commodity, then the average cost per unit is c(x) = C(x)/x. Show that if the average cost is a minimum, then the marginal cost equals the average
(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
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 + sin 3x)¹/x x→0+
Find the most general antiderivative of the function.f(t) = 2 sin t – 3et
Use the methods of this section to sketch several members of the given family of curves. What do the members have in common? How do they differ from each other?f(x) = x4 – cx, c > 0
If a and b are positive numbers, find the maximum value off(x) = xa(1 − x)b, 0 ≤ x ≤ 1.
Find the most general antiderivative of the function.f(x) = x–3 + cosh 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 (cos x)¹/x² x →0
Use the methods of this section to sketch several members of the given family of curves. What do the members have in common? How do they differ from each other?f(x) = x3 – 3c2x + 2c3, c > 0
A retailer has been selling 1200 tablet computers a week at $350 each. The marketing department estimates that an additional 80 tablets will sell each week for every $10 that the price is lowered.(a)
(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.(b) Use calculus to find the exact maximum and minimum values.f(x) = x5 − x3 + 2, −1 ≤
Use a graph to estimate the critical numbers of f(x) = |1 + 5x − x3| correct to one decimal place.
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 X 1 tan ¹x
Find the absolute maximum and absolute minimum values of f on the given interval.f(x) = 5 + 54x − 2x3, [0, 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. 1 lim x→0+ X 1 tan x
(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) –
The graph of a function f is shown. (The dashed lines indicate horizontal asymptotes.) Find each of the following for the given function g.(a) The domains of g and g'(b) The critical numbers of g(c)
Find the absolute maximum and absolute minimum values of f on the given interval.f(x) = 2x3 − 3x2 − 12x + 1, [−2, 3]
Use the guidelines of this section to sketch the curve.y = earctan x
The graph of a function f is shown. (The dashed lines indicate horizontal asymptotes.) Find each of the following for the given function g.(a) The domains of g and g'(b) The critical numbers of g(c)
(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 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 - In x) x →∞
Find the absolute maximum and absolute minimum values of f on the given interval.f(x) = x3 − 6x2 + 5, [−3, 5]
The graph of a function f is shown. (The dashed lines indicate horizontal asymptotes.) Find each of the following for the given function g.(a) The domains of g and g'(b) The critical numbers of g(c)
(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 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 (tan 2x)* x →0+
Find the absolute maximum and absolute minimum values of f on the given interval.f(x) = 3x4 − 4x3 − 12x2 + 1, [−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 xv x->0+
The graph of a function f is shown. (The dashed lines indicate horizontal asymptotes.) Find each of the following for the given function g.(a) The domains of g and g'(b) The critical numbers of g(c)
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 - 2x)¹/x x-0
Find the absolute maximum and absolute minimum values of f on the given interval.f(t) = (t2 − 4)3, [−2, 3]
(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
The graph of the velocity function of a particle is shown in the figure. Sketch the graph of a position function. DA 0
The graph of a function is shown in the figure. Make a rough sketch of an antiderivative F, given that F(0) = 1. y 0 y = f(x)
(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) –
The graph of f' is shown in the figure. Sketch the graph of f if f is continuous on [0, 3] and f (0) = –1. y 2 1 0 -1 1 y = f'(x) -0 2
(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 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 + x →∞ a X bx
(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
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 x¹/(1-x) x-1+

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