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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 limit or show that it does not exist. do r - r3 lim 2 - r2 + 3r3 r>00
Find the limit. lim tan (1/x) x>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.If f(x) > 1 for all x and limx→0 f(x) exist, then limx→0 f(x) > 1.
Use Definition 4 to find f'(a) at the given number a.f(x) = 5x4, a = -1
Find the limit or show that it does not exist. do Зx3 — 8х + 2 lim 4x3 — 5х2 — 2
Find the limit. 1 + x-1 x - 1 1 lim x? – 3x + 2
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 f is continuous at a, then f is differentiable at a.
Use Equation 5 to find f'(a) at the given number a. .2 f(x) = a = 3 x + 6'
Find the limit or show that it does not exist. do 4 - VX lim x→0 2 + X>00
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f(x) = 3x - 8
Use Equation 5 to find f'(a) at the given number a. 1 f(x) a = 1 V2x + 2
Find the limit or show that it does not exist. do (u? + 1)(2u? – 1) lim (u? + 2)? U -00
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. d²y dx? dy dx
Find f'(a).f(x) = 2x2 - 5x + 3
Find the limit or show that it does not exist. do /x+3x² lim 00-x 4х — 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.The equation x10 - 10x2 + 5 = 0 has a solution in the interval (0, 2).
Find f'(a).f(t) = t3 - 3t
Find the limit or show that it does not exist. do t + 3 lim V212 – 1 t 00
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 f is continuous at a, so is | f |.
Find f'(a).f(t) = 1/t2 + 1
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. A(p) = 4p3 + 3p
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 | f | is continuous at a, so is f.
(a) Show that f has a removable discontinuity at x = 3.(b) Redefine f(x) so that f is continuous at x = 3 (and thus the discontinuity is “removed”). x – 3 f(x) x2 – 9
Find f'(a).f(x) = x/1 - 4x
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. F(t) = t3 - 5t + 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 f is differentiable at a, so is | f |.
Find the limit or show that it does not exist. do 2x – lim x -0 Xr* + 3 4 X -00
Find the limit or show that it does not exist. do q° + 6q – 4 lim 49? - 39 + 3 9>00
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. F(v) = v/v + 2
Find the limit or show that it does not exist. do lim (/25t2 + 2 – 5t) t 00
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. g(u) = u + 1/4u - 1
If g(x) = x4 - 2, find g'(1) and use it to find an equation of the tangent line to the curve y = x4 - 2 at the point (1, -1).
Find the limit or show that it does not exist. do lim (V4x2 + 3x + 2x) X -00
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. f(x) = x4
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. 1 f(x) /1 + x
Find the limit or show that it does not exist. do lim (Va x² + ax Vx2 + bx X>00
Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative. 1 g(x) = 1 + Vx X,
Find the limit or show that it does not exist. do lim (x - Vx) X00
Find the limit or show that it does not exist. do lim (x? + 2x') 00--X
Find the limit or show that it does not exist. do 1 - e* lim 1 + 2e* X>00 x00
Find the limit or show that it does not exist. do 3x -3x e lim e 3x + e-3x + e x00
Find the limit or show that it does not exist. do lim x→(7/2)+ e sec x
Find the limit or show that it does not exist. do lim tan-(In xr) x→0+
Find the limit or show that it does not exist. do lim [In(1 + x²) - In(1 + x)] X> 00
Find the vertical asymptote of the function x - 1 f(x) = 2x + 4
Find the limit or show that it does not exist. do lim [In(2 + x) – In(1 + x)] X> 00
The graph of f is given. State, with reasons, the numbers at which f is not differentiable. yA -2 2
(a) Graph the function f(x) − ex + ln| x - 4 | for 0 ≤ x ≤ 5. Do you think the graph is an accurate representation of f?(b) How would you get a graph that represents f better?
Each limit represents the derivative of some function f at some number a. State such an f and a in each case.1 IT tan + h 1 lim h→0 h 4.
LetFind the value of c so that limt→2 B(t) exists. S4 – }t if t< 2 B(t) Vt + c if t > 2
Find the limits as x → ∞ and as x → - ∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.dy = x4 - x6
Find the limits as x → ∞ and as x → - ∞. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.dy = x3(x + 2)2(x - 1)
If a water wave with length L moves with velocity v in a body of water with depth d, thenwhere t is the acceleration due to gravity. (See Figure 5.) Explain why the approximationis appropriate in deep water.Figure 5 gL tanh 2 2nd L
The van der Waals equation for n moles of a gas iswhere P is the pressure, V is the volume, and T is the temperature of the gas. The constant R is the universal gas constant and a and b are positive constants that are characteristic of a particular gas.(a) If T remains constant, use implicit
Suppose thatf(1) = 2 f'(1) = 3 f(2) = 1 f'(2) = 2g(1) = 3 g'(1) = 1 g(2) = 1 g'(2) = 4(a) If S(x) = f (x) + g(x), find S'(1).(b) If P(x) = f(x)g(x), find P'(2).(c) If Q(x) = f(x)/g(x), find Q'(1).(d) If C(x) = f(g(x)), find
Use the formula in Exercise 83.If f (ϰ) = ϰ3 + 3 sin ϰ + 2 cos ϰ, find (f-1)'(2).
For the particle described in Exercise 7, sketch a graph of the acceleration function. When is the particle speeding up? When is it slowing down? When is it traveling at a constant speed?
If F =- f o g, where f and g are twice differentiable functions, use the Chain Rule and the Product Rule to show that the second derivative of F is given by F"(ϰ) = f"(g(ϰ)) · [g'(ϰ)]2 + f'(g(ϰ)) · g"(ϰ)
If F = f + g + h , where f, g, and h are differentiable functions, use the Chain Rule to show thatF'(ϰ) = f'(g(h(ϰ)) · g'(h(ϰ)) · h'(ϰ)
On every exponential curve y = bϰ (b > 0, b ≠ 1), there is exactly one point (ϰ0, y0) at which the tangent line to the curve passes through the origin. Show that in every case, y0 = e.
Let c be the x-intercept of the tangent line to the curve y = bϰ (b > 0, b ≠ 1) at the point (a, ba). Show that the distance between the points (a, 0) and (c, 0) is the same for all values of a.
Cobalt-60 has a half-life of 5.24 years.(a) Find the mass that remains from a 100-mg sample after 20 years.(b) How long would it take for the mass to decay to 1 mg?
Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule.Write f (ϰd/g(ϰ) = f (ϰ)[g(x)]–1.
Sketch the parabolas y = ϰ2 and y = ϰ2 – 2ϰ + 2. Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?
A tangent line is drawn to the hyperbola xy - c at a point P as shown in the figure.(a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P.(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter
Use the formula in Exercise 87 to find the derivative.(a) h(ϰ) = ϰ3(b) h(ϰ) = 3ϰ(c) h(ϰ) = (sin ϰ)ϰ
In Example 1.3.4 we arrived at a model for the length of daylight (in hours) in Philadelphia on the t th day of the year:Use this model to compare how the number of hours of daylight is increasing in Philadelphia on March 21 (t = 80) and May 21 (t = 141). 2 (t - 80) 365 L(t) = 12 + 2.8 sin
(a) Find an equation of the tangent line to the curve y = ex that is parallel to the line x − 4y = 1.(b) Find an equation of the tangent to the curve y = ex that passes through the origin.
Suppose that f and t are differentiable functions and let h(ϰ) = f(ϰ)g(ϰ). Use logarithmic differentiation to derive the formula h' = g•f9-. f' + ( Inf) fª g'
At what point on the curve y = [ln(x + 4)]2 is the tangent horizontal?
Find numbers a and b such that the given function t is differentiable at 1. Jax - 3x if x< |bx² + 2 g(x) if x>
Use the formula in Exercise 83.If f (ϰ) = ϰ + eϰ, find (f-1)'(1).
Find h' in terms of f' and g'.h(x) = f(g(sin 4x))
Use the formula in Exercise 83.If f (4) = 5 and f' (4) = 2/3, find (f-1)'(5).
Derivatives of Inverse Functions Suppose that f is a one-to-one differentiable function and its inverse function f-1 is also differentiable. Use implicit differentiation to show thatprovided that the denominator is not 0. 1 (f-)'(x) f'(f-'(x))
Find f' in terms g'.f(x) = g(In x)
Find f' in terms g'.f(x) = ln |g(x)|
If F(ϰ) = f(ϰf(ϰf(ϰ))), where f(1) = 2, f(2) = 3, f'(1) = 4, f'(2) = 5, and f'(3) = 6, find F'(1).
Find f' in terms g'.f(x) = eg(x)
Find f' in terms g'.f(x) = g(g(x))
Find the derivative of the function. Simplify where possible.y = tan-1(ϰ/a) + ln √ϰ – a/ϰ + a
Find f' in terms g'.f(x) = g[(x)]2
Let g(ϰ) = ecϰ + f(ϰ) and h(ϰ) = ekϰf(ϰ), where f(0) = 3, f'(0) = 5, and f"(0) = –2.(a) Find t'(0) and g"(0) in terms of c.(b) In terms of k, find an equation of the tangent line to the graph of h at the point where ϰ = 0.
Find the derivative of the function. Simplify where possible.y = cos–1(sin–1 t)
LetIs f differentiable at 1? Sketch the graphs of f and f'. x? + 1 if x 1
Find f' in terms g'.f(x) = x2g(x)
Find a parabola with equation y = aϰ2 + bϰ + c that has slope 4 at ϰ = 1, slope –8 at ϰ = –1, and passes through the point (2, 15).
Find the derivative of the function. Simplify where possible.h(t) = cot–1(t) + cot–1(1/t)
The equation y" + y' – 2y = ϰ2 is called a differential equation because it involves an unknown function y and its derivatives y' and y". Find constants A, B, and C such that the function y = Aϰ2 + Bϰ + C satisfies this equation. (Differential equations will be studied in detail in Chapter 9.)
Find the derivative of the function. Simplify where possible.y = tan–1(ϰ –√1 + ϰ2)
If f and t are the functions whose graphs are shown, let u(ϰ) = f(g(ϰ)), v(ϰ) = g(f (ϰ)), and w(ϰ) = g(g(ϰ)). Find each derivative, if it exists. If it does not exist, explain why.(a) u'(1)(b) v'(1)(c) w'(1) y. f 1
Find the derivative of the function. Simplify where possible.f (z) = earcsin(z2)
If f(x) = (x − a)(x − b)(x − c), show that f'(x) f(x) 1 1 1 x - a x - b x - c
Find the derivative of the function. Simplify where possible.g(t) = ln(arctan(t4))
Find the derivative of the function. Simplify where possible.h(ϰ) = (arcsin ϰ) ln ϰ
(a) Find equations of both lines through the point (2, –3) that are tangent to the parabola y = ϰ2 + ϰ.(b) Show that there is no line through the point (2, 7) that is tangent to the parabola. Then draw a diagram to see why.
Let f (ϰ)(a) Graph f . What type of discontinuity does it appear to have at 0?(b) Calculate the left and right limits of f at 0. Do these values confirm your answer to part (a)? /1 - cos 2x
(a) If f(x) = 4x − tan x, −π/2 < x < π/2, find f' and f".(b) Check to see that your answers to part (a) are reasonable by comparing the graphs of f , f', and f".
If F(ϰ) = f (g(ϰ)), where f (–2) = 8, f' (–2) = 4, f' (5) = 3, g(5) = –2, and g'(5) = 6, find F'(5).
Find the derivative of the function. Simplify where possible.y = (tan–1ϰ)2

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