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mathematics
precalculus

Calculus Early Transcendentals 8th edition James Stewart - Solutions

Find the limit. Vx? – 9 lim — 6 x-0 2x
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 has domain [0, ∞] and has no horizontal asymptote, then limx → ∞ f (x) = ∞ or limx → ∞ f (x) = -∞.
The graph (from the US Department of Energy) shows how driving speed affects gas mileage. Fuel economy F is measured in miles per gallon and speed v is measured in miles per hour.(a) What is the meaning of the derivative F′(v)?(b) Sketch the graph of F′(v).(c) At what speed should you drive if
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = 3x* – 5x + x? + 4, a = 2
Find the limit. x² - 9 lim х>0 2х — 6
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. p(v) = 2/3v² + 1, a=1
Evaluate the limit and justify each step by indicating the appropriate properties of limits. 19x3 + 8х — 4 lim х> оо Уз — 5х + х3
Evaluate the limit and justify each step by indicating the appropriate properties of limits. 2x2 – 7 lim 5x? + x – 3
Calculate y'.y = √sin √x
Evaluate the limit, if it exists. x² + 3x lim х2 — х — 12 |х>-3 х*
Find the limit. Vx + 6 - x lim .3 x' - 3x?
Find the derivative of the function.s(t) = √θ1 + sin t/1 + cos t
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 p is a polynomial, then limx → b p(x) = p(b).
Find the limit. ut – 1 и lim и>1 из + 5и? — 6и
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 limx → 6 [f (x) g(x)] exists, then the limit must be f (6) g(6).
Find the limit. 4 - v lim v→4+ |4 – v|
Differentiate.f(x) = ax + b/cx + d
Given that limx →π csc2 x − ∞, illustrate Definition 6 by finding values of δ that correspond to (a) M = 500 and (b) M = 1000.
Differentiate the function.k(r) = er + re
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 limx → a f (x) exists but limx → a g(x) does not exist, then limx → a [f (x) + g(x)] does not exist.
(a) Find an equation of the tangent line to the curve y = 3x + 6 cos x at the point (π/3, π + 3).(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) Use a graph to find a number δ such that if 2 < x < 2 + δ then 1/in(x - 1) > 100(b) What limit does part (a) suggest is true?
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 neither limx → a f (x) nor limx → a tsxd exists, then limx → a [f (x) + g(x)] does not exist.
Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f′ below it. х
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). 12х? + 1 lim х—2 У 3х — 2
Use implicit differentiation to find an equation of the tangent line to the curve at the given point.sin(x + y) = 2x - 2y, (π, π)
Find the limit. lim r-9 9 (r – 9)*
If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height (in meters) after t seconds is given by H = 10t - 1.86t2.(a) Find the velocity of the rock after one second.(b) Find the velocity of the rock when t = a.(c) When will the rock hit the surface?(d) With what velocity
Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places).
The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 2 sin πt 1 3 cos πt, where t is measured in seconds.(a) Find the average velocity during each time period:(i) [1, 2] (ii) [1, 1.1](iii) [1, 1.01] (iv) [1,
Use a linear approximation (or differentials) to estimate the given number.√100.5
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). 12 – 2 lim t - 3t + 5 t-2
Find the limit. t? – 4 lim -2 t - 8
Shown are graphs of the position functions of two runners, A and B, who run a 100-meter race and finish in a tie.(a) Describe and compare how the runners run the race.(b) At what time is the distance between the runners the greatest?(c) At what time do they have the same velocity? s (meters), A 80
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). lim (1 + x)(2 – 6x² + x')
Calculate y'.sin(xy) = x2 - y
Find an equation of the tangent line to the curve at the given point.
Find the limit. (h – 1) + 1 lim
Use the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain why.(a) (b)(c) lim f(x) lim f(x) x-
The table shows the position of a motorcyclist after accelerating from rest.(a) Find the average velocity for each time period:(i) [2, 4} (ii) [3, 4] (iii) [4, 5] (iv) [4, 6](b) Use the graph of s as a function of t to estimate the instantaneous velocity when t − 3. 01 2 3 4.9 20.6 46.5 79.2
Find the derivative of the function.g(u) = (u3 - 1/u3 + 1)8
Find an equation of the tangent line to the curve at the given point.y = √x , (1, 1)
Evaluate sin(a + 2x) – 2 sin(a + x) + sin a lim x?
Differentiate the function.f (x) = 240
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.
Find an equation of the tangent line to the curve at the given point.y = x3 - 3x + 1, (2, 3)
Differentiate.f (x) = x/x + c/x
Differentiate the function.j(x) = x2.4 + e2.4
Sketch the graph of the function and use it to determine the values of a for which limx → a f (x) exists. 1 + sin x if x < 0 if 0 T
If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters t seconds later is given by y = 10t - 1.86t2.(a) Find the average velocity over the given time intervals:(i) [1, 2] (ii) [1, 1.5](iii) [1, 1.1] (iv) [1, 1.01](v) [1, 1.001](b) Estimate the
Calculate y'.y = 1/3√x + √x
Find the limit. x? – 9 lim 1+ x? + 2x – 3
Evaluate 12х — 1| - 12х +1| lim х
Trace or copy the graph of the given function f. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of f′ below it. y. х
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). u* + Зи + 6 lim yu 6 и—-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 limx→5 f (x) − 0 and limx→5 g(x) − 0, then limx→5 [f (x)/g(x)] does not exist.
Find the derivative of the function.f(t) = 2t3
Find an equation of the tangent line to the curve at the given point.y = x + tan x, (π, π)
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. lim (x – 3) lim lim (x? + 2x – 4) + 2x - 4 -l x
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. lim х? + 6х —7 (х + 6х — 7) lim нx + 5х - 6 lim (x? + 5x – 6)
If f and t are differentiable functions with f (0) = g(0) = 0 and t'(0) ± 0, show that f(x) lim f'(0) g(x) (0),6
Evaluate х lim x→1 х
Differentiate.F(t) = At/Bt2 + Ct3
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. 2х 2х lim lim 4 4 х — 4 lim 4х — 4 х — 4 х — 4
A particle moves along the curve y = 2 sin(x/2). As the particle passes through the point (13, 1), its x-coordinate increases at a rate of s10 cm/s. How fast is the distance from the particle to the origin changing at this instant?
The graphs of f and t are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why.(a)(b)(c)(d)(e)(f) lim [f(x) + g(x)] lim [f(x) g(x)] –
Differentiate the function. 17 G(t) = 5t +
Use the given graph to estimate the value of each derivative. Then sketch the graph of f 9.(a) f′(0) (b) f′(1) (c) f′(2) (d) f′(3)(e) f′(4) (f) f′(5)(g) f′(6) (h) f′(7) 1 1
Find the limit. 3. lim e
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). t* – 2 lim --2 2t2 – 3t + 2
Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy.y sec x = x tan y
Find the limit. lim .2 x² + 2x – 3 x-3 3
(a) The figure shows an isosceles triangle ABC with ∠B − ∠C. The bisector of angle B intersects the side AC at the point P. Suppose that the base BC remains fixed but the altitude |AM| of the triangle approaches 0, so A approaches the midpoint M of BC. What happens to P during this process?
Use a linear approximation (or differentials) to estimate the given number.1/4.002
Use the definitions of the hyperbolic functions to find each of the following limits. (a) lim tanh x (b) lim tanh x (c) lim sinh x (d) lim sinh x (f) lim coth x (e) lim sech x (g) lim coth x (h) lim coth x sinh x (j) lim e* (i) lim csch x
Sketch the graph of a function f that is continuous except for the stated discontinuity.Neither left nor right continuous at -2, continuous only from the left at 2
Calculate y'.y = (1 - x2 + )-1
Find the derivative of the function.y = etan θ
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 limx → 5 f (x) = 2 and limx → 5 g(x) = 0, then limx → 5 [f (x)/g(x)] does not exist.
Find an equation of the tangent line to the curve at the given point.y = cos x - sin x, (π, -1)
Sketch the graph of a function f that is continuous except for the stated discontinuity. Discontinuities at -1 and 4, but continuous from the left at -1 and from the right at 4
Evaluate the following limits, if they exist, where v x b denotes the greatest integer function.(a)(b) [x] lim lim x [1/x] X-
From the graph of g, state the intervals on which g is continuous. y. + х -3 -2 1 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. .2 x? - 9 lim x-3 x - 3 lim (x + 3)
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. x? – 9 — х + 3 х — 3
Suppose that we replace the parabolic mirror of Problem 22 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle with center O. A ray of light coming in toward the mirror parallel to the axis along the line PQ
Differentiate.f (x) = x2ex/x2 + ex
Differentiate the function. х? + 4x + 3 y = х
Compute Δy and dy for the given values of x and dx = Δx. Then sketch a diagram like Figure 5 showing the linesegments with lengths dx, dy, and Δy.y = ex, x = 0, Δx = 0.5
Calculate y'.y = sec(1 + x2)
Find the derivative of the function.y = (x + 1/x)5
Find an equation of the tangent line to the curve at the given point.y = ex cos x, (0, 1)
Differentiate.V(t) = 4 - t/tet
If g(x) + x sin g(x) = x2, find g'(0).
Differentiate the function. Vx + x y =
Use the given graph of f to state the value of each quantity, if it exists. If it does not exist, explain why.(a)
Compute Δy and dy for the given values of x and dx = Δx. Then sketch a diagram like Figure 5 showing the linesegments with lengths dx, dy, and Δy.y = √x - 2, x = 3, Δx = 0.8
The point P(0.5, 0) lies on the curve y − cos πx.(a) If Q is the point (x, cos πx), use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the following values of x:(i) 0 (ii) 0.4 (iii) 0.49(iv) 0.499 (v) 1 (vi) 0.6(vii) 0.51 (viii)
Calculate y'.y = 3x In x
Find the limit. x? – 9 lim x-3 x? + 2x – 3

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