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study help
mathematics
precalculus
Calculus Early Transcendentals 8th edition James Stewart - Solutions
Investigate the family of functionsfn(x) = tanh(n sin x)where n is a positive integer. Describe what happens to the graph of fn when n becomes large.
(a) Prove thatandif these limits exist.(b) Use part (a) and Exercise 65 to find lim f(x) = lim f(1/t) lim f(x) = lim f(1/t)
Evaluate /1 + tan x lim - V1 + sin x .3 x³
Express the limit as a derivative and evaluate. cos 0 – 0.5 lim 0 – T/3 0→ T/3 1/3
Express the limit as a derivative and evaluate. V16 + h – 2 lim
Express the limit as a derivative and evaluate. x17 - lim lim
Evaluate dy if y = x3 - 2x2 + 1, x = 2, and dx = 0.2.
If y = f (u) and u = t(x), where f and t possess third derivatives, find a formula for d3y/dx3 similar to the one given in Exercise 99.
Prove that f is continuous at a if and only if lim f(a + h) = f (a)
A cup of hot chocolate has temperature 80°C in a room kept at 20°C. After half an hour the hot chocolate cools to 60°C.(a) What is the temperature of the chocolate after another half hour?(b) When will the chocolate have cooled to 40°C?
Prove, without graphing, that the graph of the function has at least two x-intercepts in the specified interval. y = x? – 3 + 1/x, (0, 2)
For the limitillustrate Definition 9 by finding a value of N that corresponds to M − 100. = 00 lim Vx In x
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?
For the limitillustrate Definition 8 by finding values of N that correspond to ε − 0.1 and ε − 0.05. 1 - 3x lim Vx² + 1 = 3
Prove, without graphing, that the graph of the function has at least two x-intercepts in the specified interval. y = sin x, (1, 2)
For the limitillustrate Definition 7 by finding values of N that correspond to ε − 0.1 and ε − 0.05. 1 - 3x lim Vx2 + 1 -3
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.y = x2 (x2 - 1)2 (x + 2)
(a) Prove that the equation has at least one real root.(b) Use your graphing device to find the root correct to three decimal places.arctan x = 1 - x
The volume of a right circular cone is V = 1/3π r2h, where r is the radius of the base and h is the height.(a) Find the rate of change of the volume with respect to the height if the radius is constant.(b) Find the rate of change of the volume with respect to the radius if the height is constant.
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.y = x3(x + 2) (x - 1)
(a) Prove that the equation has at least one real root.(b) Use your calculator to find an interval of length 0.01 that contains a root. In x = 3 – 2.x
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. sin x 3D х? — х, (1,2)
A particle moves along a horizontal line so that its coordinate at time t is x = √b2 + c2t2 , t > 0, where b and c are positive constants.(a) Find the velocity and acceleration functions.(b) Show that the particle always moves in the positive direction.
Air is being pumped into a spherical weather balloon. At any time t, the volume of the balloon is V(t) and its radius is r(t).(a) What do the derivatives dV/dr and dV/dt represent?(b) Express dV/dt in terms of dr/dt.
A particle moves along a straight line with displacement s(t), velocity v(t), and acceleration a(t). Show that a(t) = v(t) dv/dsExplain the difference between the meanings of the derivatives dv/dt and dv/ds.
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.y = x4 - x6
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. e* — 3 — 2х, (0, 1) it
Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. - Jx, (2, 3) In x = x
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.y = 2x3 - x4
The function C(t) = K(e-at - e-bt), where a, b, and K are positive constants and b > a, is used to model the concentration at time t of a drug injected into the bloodstream.(a) Show that limt→∞ C(t) = 0.(b) Find C'(t), the rate of change of drug concentration in the blood.(c) When is this
Prove thatUse lim Vx = Va if a > 0. х x + Ja
Let P and Q be polynomials. Find if the degree of P is (a) Less than the degree of Q and(b) Greater than the degree of Q. P(x) lim Q(x) х-
Sketch the parabolas y = x2 and y = x2 - 2x + 2. Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?
In Section 1.4 we modeled the world population from 1900 to 2010 with the exponential functionWhere t = 0 corresponds to the year 1900 and P(t) is measured in millions. According to this model, what was the rate of increase of world population in 1920? In 1950? In 2000? P(t) = (1436.53) ·
Prove that lim х—2 х
The average blood alcohol concentration (BAC) of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration functionC(t) − 0.0225te-0.0467tWhere t is measured in minutes after consumption and
Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola y = x2. Where do these lines intersect?
Evaluate 1000 х - 1 lim х — 1 х>1
A tangent line is drawn to the hyperbola xy = c at a point P.(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 where P is located on
(a) Graph the function(b) By calculating values of f (x), give numerical estimates of the limits in part (a).(c) Calculate the exact values of the limits in part (a). Did you get the same value or different values for these two limits? [In view of your answer to part (a), you might have to check
Find h' in terms of f' and t'.h(x) = f(t(sin 4x))
Find h' in terms of f' and t'.h(x) = √ f(x)/g(x)
The graph of any quadratic function f (x) − ax2 + bx + c is a parabola. Prove that the average of the slopes of the tangent lines to the parabola at the endpoints of any interval [p, q] equals the slope of the tangent line at the midpoint of the interval.
Find h' in terms of f' and t'.h(x) = f (x) t(x) f (x) 1 t(x) 80.
What is the value of c such that the line y = 2x + 3 is tangent to the parabola y = cx2?
Evaluate /6 — х — 2 lim (3 — х — 1 - 2 х—2
Find f' in terms of g'.f(x) = t (ln x)
Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. 2e* e* – 5
Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. х3. — х y = х? — 6х + 5
If prove that limx → 0 f (x) − 0. .2 x? if x is rational if x is irrational F(x) =
If find the following limits.(a)(b) f(x) - 5, lim .2 х* lim f(x)
(a) Show that f (x) = x + ex is one-to-one.(b) What is the value of f-1(1)?(c) Use the formula from Exercise 77(a) to find s f-1)'(1).
If r is a rational function, use Exercise 57 to show that limx → a r(x) − r(a) for every number a in the domain of r.Exercise 57If p is a polynomial, show that limx → a p(x) − p(a).
Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. 1 + x* y .2 - x4
Determine the infinite limit. x csc x lim х>2п
Let P represent the percentage of a city’s electrical power that is produced by solar panels t years after January 1, 2000.(a) What does dP/dt represent in this context?(b) Interpret the statement dP = 3.5 dt t=2
Find the value of c such that the line y − 3/2 x + 6 is tangent to the curve y = c√x.
Find the 1000th derivative of f(x) = xe-x.
Let f (x) − [[cos x ]], -π ≤ x ≤ π.(a) Sketch the graph of f.(b) Evaluate each limit, if it exists.(i)(ii)(iii)(iv)(c) For what values of a does limx→xaf(x) exist? lim f(x) х- lim f(x) x>(7/2)-
Find f' in terms of g'.f(x) = ln |t(x)|
Find f' in terms of g'.f(x) = et(x)
Suppose the curve y = x4 + ax3 + bx2 + cx + d has a tangent line when x − 0 with equation y = 2x + 1 and a tangent line when x = 1 with equation y = 2 - 3x. Find the values of a, b, c, and d.
For what values of r does the function y = erx satisfy the differential equation y'' - 4y' + y = 0?
Find f' in terms of g'.f(x) = g(ex)
Find f' in terms of g'.f(x) = g (g (x)
Find the limit or show that it does not exist. 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. y. -2 4 х
Let g(x) − sgn(sin x).(a) Find each of the following limits or explain why it does not exist.(i) (ii)(iii)(iv)(v)(vi)(b) For which values of a does limx → a g(x) not exist?(c) Sketch a graph of t. lim g(x) x→0+ lim g(x) х>0-
Where is the function h (x) = |x - 1| + |x + 2| differentiable? Give a formula for h′ and sketch the graphs of h and h′.
If F(x) = f (x f (x f (x))), where f (1) = 2, f (2) = 3, f' (1) = 4, f' (2) = 5, and f' (3) = 6, find F' (1).
Find f' in terms of g'.f(x) = [g(x)]2
Find f' in terms of g'.f(x) = g(x2)
(a) Use implicit differentiation to find y' ifx2 + xy + y2 + 1 = 0(b) Plot the curve in part (a). What do you see? Prove that what you see is correct.(c) In view of part (b), what can you say about the expression for y' that you found in part (a)?
Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. 2x? + x – 1 У х2 +x — 2 х* 2
Suppose f is continuous on [1, 5] and the only solutions of the equation f (x) − 6 are x − 1 and x − 4. If f (2) − 8, explain why f (3) > 6.
Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. 2x² + 1 y = Зx? + 2х — 1
Nick starts jogging and runs faster and faster for 3 mintues, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes.(a) Sketch a possible graph of the distance s Nick has covered after t minutes.(b) Sketch a graph of
Prove that lim In x x→0+ = -0.
Find the limit, if it exists. If the limit does not exist, explain why. 2х — 1 lim х—0.5— |2x³ – x²| х>0.5- |2х3 — х?
Determine the infinite limit. lim Inx
Use the definition of a derivative to find f′(x) and f′(x). Then graph f, f′, and f′′ on a common screen and check to see if your answers are reasonable.f (x) = x3 - 3x
Consider the function (a) Show that(b) Show that(c) What can you conclude about f(x) = tan |f(x) = 0 for x п 2т 3т
Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. 5 + 4x y = x + 3
Determine the infinite limit. x? – 2x – 8 lim x-2+ x? – 5x + 6
Let f (x) = 1/x and g(x) − 1/x2.(a) Find (f° g) (x).(b) Is f° g continuous everywhere? Explain.
Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. 2 if x
Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. x if x
Find the limit or show that it does not exist. lim [In(1 + x?) – In(1 + x)] X 00
Show that f is continuous on (-∞, ∞). |sin x if x < /4 [ sin f(x) = cos x if x > /4 %3D >T/4
Find the limit or show that it does not exist. lim tan-(In x)
At what numbers is the following function t differentiable?Give a formula for g′ and sketch the graphs of g and g′. 2x if x < 0 if 0 < x < 2 g(x) = {2x – x² if x > 2
Determine the infinite limit. x? – 2x .2 lim x→2- x² – 4x + 4
Suppose N is the number of people in the United States who travel by car to another state for a vacation this year when the average price of gasoline is p dollars per gallon. Do you expect dN/dp to be positive or negative? Explain.
Find the limit, if it exists. If the limit does not exist, explain why. 2x + 12 lim x→-6 x + 6
If t is a twice differentiable function and f (x) − xg(x2), find f'' in terms of g, g', and g''.
Find the limit or show that it does not exist. lim (e-2* cos x)
Show that f is continuous on (-∞, ∞). |1 – x² if x < 1 if x>1 f(x) Inx
Find f' in terms of g'.f (x) = x2g(x)
Let r(x) = f (g(h(x))), where h(1) = 2, t(2) = 3, h'(1) = 4, t'(2) = 5, and f' (3) − 6. Find r'(1).
Prove that lim Vx esin(7/x) = 0.
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![lim [In(2 + x) – In(1 + x)] X 00](https://dsd5zvtm8ll6.cloudfront.net/si.question.images/images/question_images/1547/5/7/0/1575c3e0bedd9de41547552807405.jpg){kind=small}
![lim [In(1 + x?) – In(1 + x)] X 00](https://dsd5zvtm8ll6.cloudfront.net/si.question.images/images/question_images/1547/5/7/0/0855c3e0ba500fd71547552734518.jpg){kind=small}
