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mathematics
calculus early transcendentals 9th

Questions and Answers of Calculus Early Transcendentals 9th

The figure shows a lamp located three units to the right of the y-axis and a shadow created by the elliptical region ϰ2 + 4y2 ≤ 5. If the point (–5, 0) is on the edge of the shadow, how far
(a) If f(x) = x√5 − x , find f'(x).(b) Find equations of the tangent lines to the curve y = x√5 − x at the points (1, 2) and (4, 4).(c) Illustrate part (b) by graphing the curve and tangent
Find the derivative of the function. Simplify where possible.y = tan–1(ϰ2)
Find the derivative of the function. Simplify where possible.y = tan–1√ϰ – 1
Use implicit differentiation to find dy/dx for the equationϰ/y = y2 + 1 y ≠ 0and for the equivalent equationϰ = y3 + y y ≠ 0Show that although the expressions you get for dy/dx look different,
The function f (ϰ) = sin(ϰ + sin 2ϰ), 0 ≤ x ≤ π, arises in applications to frequency modulation (FM) synthesis.(a) Use a graph of f produced by a calculator or computer to make a rough sketch
Find the derivative of the function. Simplify where possible.g(ϰ) = sec–1(eϰ)
(a) Evaluate(b) Evaluate(c) Illustrate parts (a) and (b) by graphing y = ϰ sin(1/ϰ). lim x sin . X SI
Find equations for two lines that are both tangent to the curve y = ϰ3 – 3ϰ2 + 3ϰ – 3 and parallel to the line 3ϰ – y = 15.
Extended Product Rule The Product Rule can be extended to the product of three functions.(a) Use the Product Rule twice to prove that if f, g, and h are differentiable, then (fgh)' = f'gh + fg'h +
(a) If f (ϰ) = ϰ√2 – ϰ2 , find f'(ϰ).(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f'.
Find the derivative of the function. Simplify where possible.f(ϰ) = sin–1(5ϰ)
Find an equation of the line that is both tangent to the curve y = ϰ4 + 1 and parallel to the line 32ϰ – y = 15.
(a) The curve y = |x|/√2 – ϰ2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1).(b) Illustrate part (a) by graphing the curve and the tangent
Show that lim (1 + e* for any x > 0. n
Find the limit. sin(x – 1) lim x-1 x2 + x - 2
Find an equation of the tangent line to the curve at the given point.y = ϰe–ϰ 2, (0, 0)
Find d9/dϰ9 (ϰ8 ln ϰ).
Find the limit. 1- tan x lim xT/4 sin x - cos x
Find the points on the curve y = ϰ3 + 3ϰ2 – 9ϰ + 10 where the tangent is horizontal.
In this exercise we estimate the rate at which the total personal income is rising in Boulder, Colorado. In 2015, the population of this city was 107,350 and the population was increasing by roughly
Find an equation of the tangent line to the curve at the given point.y = sin(sin ϰ), (π, 0)
Find a formula for f(n)(ϰ) if f (ϰ) = ln(ϰ – 1).
Find f(n)(x) if f(x) = 1/(2 − x).
Find the limit. sin(x?) lim
Use the method of Exercise 57 to compute Q'(0), where 1+х + x? + хе* 1 — х + x? — хе* Q(x) .2 xe*
Find an equation of the tangent line to the curve at the given point.y = √1 + ϰ3 , (2, 3)
Find y' if ϰy = yϰ.
Find y" if x6 + y6 = 1.
Find the limit. cos e - 1 lim cos 202
Find an equation of the tangent line to the curve at the given point.y = 2ϰ, (0, 1)
If g(θ) = θ sin θ, find g"(π/6).
Find y' and y".y = ee ϰ
Orthogonal Trajectories Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each
Use logarithmic differentiation to find the derivative of the function.y = (ln ϰ)cos ϰ
Find the limit. lim csc x sin(sin x)
Find y' and y".y = √cos ϰ
Orthogonal Trajectories Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each
Use logarithmic differentiation to find the derivative of the function.y = ϰln ϰ
Find the limit. sin e lim 0 0 + tan 0
Calculate y'.y = sin2(cos√sinπx)
Find y' and y".y = (1 + √ϰ)3
Orthogonal Trajectories Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each
Use logarithmic differentiation to find the derivative of the function.y = (sin ϰ)ln ϰ
Find the limit. sin 3x sin 5x lim x?
Calculate y'.y = cos(e√tan 3x)
Find the derivative. Simplify where possible. y = x sinh (x/3) – 19 + x2
Find y' and y".y = cos(sin 3θ)
Orthogonal Trajectories Two curves are orthogonal if their tangent lines are perpendicular at each point of intersection. Show that the given families of curves are orthogonal trajectories of each
Use logarithmic differentiation to find the derivative of the function.y = (cos ϰ)ϰ
Find the limit. sin 3x lim .3 -0 5x - 4x
Suppose that the volume V of a rolling snowball increases so that dV/dt is proportional to the surface area of the snowball at time t. Show that the radius r increases at a constant rate, that is,
Calculate y'.y = x tanh-1√x
Find the derivative. Simplify where possible. y = x tanhx + In/1 – x2
Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f , f', and f"f (ϰ) = eϰ – ϰ3
Find the derivative of the function.y = sin3(cos(ϰ2))
Use logarithmic differentiation to find the derivative of the function.y = (√ϰ)ϰ
Find the limit. sir lim 0-0 tan 70
Find the derivative. Simplify where possible. G(u) 3 сosh 'V1 + u?, и > 0
If f and t are the functions whose graphs are shown, let u(ϰ) = f (ϰ)g(ϰ) and v(ϰ) = f (ϰ)/g(ϰ).(a) Find u'(1).(b) Find v'(4). y. g 1 1
Find the derivative of the function.y = cos√sin(tan πϰ)
Use logarithmic differentiation to find the derivative of the function.y = ϰsin ϰ
Find the limit. tan 2x lim
Calculate y'. x? - 4 y = In 2x + 5
Find the derivative. Simplify where possible.y = sech-1(sin θ), 0 < θ < π/2
In physics textbooks, the period T of a pendulum of length L is often given as T ≈ 2π√L/g , provided that the pendulum swings through a relatively small arc. In the course of deriving this
Find the first and second derivatives of the function.G(r) = √r + 3√r
Find the derivative of the function.y = sin(θ + tan(θ + cos θ))
Use logarithmic differentiation to find the derivative of the function.y = ϰ1/ϰ
Find the limit. 1- sec x lim 2x
If f(2) = 10 and f'(ϰ) = ϰ2 f(ϰ) for all ϰ, find f"(2).
Find the first and second derivatives of the function.f (x) = 0.001ϰ5 – 0.02ϰ3
Find the derivative of the function.y = (3cos(ϰ2) – 1)4
Find the limit. sin x - sin x cos x lim .2
If g(ϰ) = ϰf(ϰ), where f(3) = 4 and f'(3) = –2, find an equation of the tangent line to the graph of t at the point where ϰ = 3.
Calculate y'. sin mx y =
Find the derivative. Simplify where possible.g(x) = tanh-1(x3)
Find the derivative of the function. y= 234*
Find the limit. sin? 3x lim 『ー
Calculate y'.y = x sinh(x2)
Find the derivative. Simplify where possible.f(x) = sinh-1(−2x)
Find the derivative of the function.f (ϰ) = esin2(ϰ2)
Find the limit. sin 3t lim 0 sin t
Calculate y'. (х + 1)* y = x* + A4
Find the derivative. Simplify where possible. 1 + sinh t f(t) 1- sinh t
Find the derivative of the function.y = √ϰ + √ϰ + ǀ√ϰ
Find the limit. sin x lim x0 sin 7x
Calculate y'.
Find the derivative. Simplify where possible. g(t) = t coth /t? + 1
Use logarithmic differentiation to find the derivative of the function.y = (ϰ2 + 2)2(ϰ4 + 4)4
Find the limit. sin 5x lim x 0 3x
Calculate y'.y + ln y = xy2
An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle θ with the plane, then the magnitude of the force isF =
Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.y = x – √x , (1, 0)
Find the derivative. Simplify where possible.y = sech(tanh x)
If g(ϰ) = ϰ/eϰ, find g(n)(ϰ).
If ϰ2 + ϰy + y3 = 1, find the value of y"' at the point where ϰ = 1.
Find the derivative of the function.f (t) = e1/t√t2 – 1
Let f (ϰ) = logb(3ϰ2 – 2). For what value of b is f' (1) = 3?
The difficulty of “acquiring a target” (such as using a mouse to click on an icon on a computer screen) depends on the ratio between the distance D to the target and width W of the target.

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