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Biocalculus Calculus Probability And Statistics For The Life Sciences 1st Edition James Stewart, Troy Day - Solutions

Find the limit. lim x+6-x x-3x-3x
Find the limit. lim u1 u1u3+5u2 - 6u
Find the limit. lim 14+ 4-v |4-v|
Find the limit. lim 7-9 (r - 9)4
Find the limit. t - 4 1-2 13-8 lim
Find the limit. lim h0 (h-1) +1 h
Find the limit. x-9 lim x1+x + 2x - 3
Find the limit. lim x-9 x-3x + 2x - 3
Find the limit. lim x-9 x-3x + 2x - 3
Find the limit. 3. lim ex-x x1 I
Find the limit. lim 3-21 100
Find the limit. lim 1-x
Sketch the graph of an example of a function f that satisfies all of the following conditions:lim x l 2`f sxd − 22, lim x l `f sxd − 0, lim x l 23 f sxd − `, lim x l 32 f sxd − 2`, lim x l 31 f sxd − 2, f is continuous from the right at 3
The graph of f is given.(a) Find each limit, or explain why it does not exist.(i) lim xl21 f sxd (ii) lim xl231 f sxd(iii) lim xl23 f sxd (iv) lim xl4 f sxd(v) lim xl0 f sxd (vi) lim xl22 f sxd(vii) lim xl`f sxd (viii) lim xl2`f sxd(b) State the equations of the horizontal asymptotes.(c) State the
Logistic equation Plot enough terms of the logistic difference equation to see the behavior of the terms. If the sequence appears to be convergent, estimate its limit and then, assuming the limit exists, find its exact value. If not, describe how the terms behave.(a) xt11 − 2.5xts1 2 xtd, x0 −
Express the repeating decimal 1.2345345345 . . . as a fraction.
Drug concentration A patient is injected with a drug at the same time every day. Before each injection, the concentration of the drug has dropped to 20% of its original value and the new dose raises the concentration by 0.25 mgymL.(a) What is the concentration after four doses?(b) If Cn is the
Calculate the first eight terms of the sequence defined by a1 − 1, an11 − 13 an 1 3. Does it appear to be convergent?Assuming the limit exists, find its exact value.
Determine whether the sequence is convergent or divergent.If it is convergent, find its limit. (-2)" (7-)="v
Determine whether the sequence is convergent or divergent.If it is convergent, find its limit. an n 1 +n 2
Determine whether the sequence is convergent or divergent.If it is convergent, find its limit. an 9n+1 10"
Determine whether the sequence is convergent or divergent.If it is convergent, find its limit. an 3 2 + n 1 + 2n
If f is continuous on f21, 1g and f s21d − 4 and f s1d − 3, then there exists a number r such that |r | , 1 and f srd − .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 5 and f s5d − 2 and f s4d − 3, then limx l 2 f s4x2 2 11d − 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 s1d . 0 and f s3d , 0, then there exists a number c between 1 and 3 such that f scd − 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 the line x − 1 is a vertical asymptote of y − f sxd, then f is not defined at 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 has domain f0, `d and has no horizontal asymptote, then limx l ` f sxd − ` or limx l ` f sxd − 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.
A function can have two different horizontal asymptotes.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 l 0 f sxd − ` and limx l 0 tsxd − `, then limx l 0 f f sxd 2 tsxdg − 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 p is a polynomial, then limx l b psxd − psbd.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 l 6 f f sxd tsxdg exists, then the limit must be f s6d ts6d.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 limxl5 f sxd − 0 and limx l 5 tsxd − 0, then limx l 5 f f sxdytsxdg does not exist.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 l 5 f sxd − 2 and limx l 5 tsxd − 0, then limx l 5 f f sxdytsxdg does not exist.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 l 1 x 2 3 x2 1 2x 2 4−lim x l 1 sx 2 3d lim x l 1 sx2 1 2x 2 4d 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 l 1 x2 1 6x 2 7 x2 1 5x 2 6−lim x l 1 sx2 1 6x 2 7d lim x l 1 sx2 1 5x 2 6d 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 l 4 S 2x x 2 4 28 x 2 4D− lim x l 4 2x x 2 4 2 lim x l 4 8x 2 4 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.
0.99999 . . . − 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 limnl` an − L, then limnl` a2n11 − L.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.
What does the Intermediate Value Theorem say?
(a) What does it mean for f to be continuous at a?(b) What does it mean for f to be continuous on the interval s2`, `d? What can you say about the graph of such a function?
Which of the following curves have vertical asymptotes?Which have horizontal asymptotes?(a) y − x4 (b) y − sin x(c) y − tan x (d) y − ex(e) y − ln x (f) y − 1yx(g) y − sx
(a) What does it mean to say that the line x − a is a vertical asymptote of the curve y − f sxd? Draw curves to illustrate the various possibilities.(b) What does it mean to say that the line y − L is a horizontal asymptote of the curve y − f sxd? Draw curves to illustrate the various
What does the Squeeze Theorem say?
State the following Limit Laws for functions.(a) Sum Law (b) Difference Law(c) Constant Multiple Law (d) Product Law(e) Quotient Law (f) Power Law(g) Root Law
Explain what each of the following means and illustrate with a sketch.(a) lim xla f sxd − L (b) lim xla1 f sxd − L(c) lim xla2 f sxd − L (d) lim xla f sxd − `(e) lim xl`f sxd − L
(a) What is the sum of the finite geometric series a 1 ar 1 ar 2 1 • • • 1 ar n?(b) If 21 , r , 1, what is the sum of the infinite geometric series a 1 ar 1 ar 2 1 • • • 1 ar n 1 • • •?
What is lim nl`r n in the following three cases?(a) 0 , r , 1 (b) r − 1 (c) r . 1
(a) What is a convergent sequence?(b) What does limnl` an − 3 mean?
A Tibetan monk leaves the monastery at 7:00 am and takes his usual path to the top of the mountain, arriving at 7:00 pm. The following morning, he starts at 7:00 am at the top and takes the same path back, arriving at the monastery at 7:00 pm. Use the Intermediate Value Theorem to show that there
Show that the function f sxd − Hx4 sins1yxd 0if x ± 0 if x − 0 is continuous on s2`, `d.
Is there a number that is exactly 1 more than its cube?
sx 2 5 −1 x 1 3(a) Prove that the equation has at least one real solution.(b) Use your graphing device to find the solution correct to three decimal places.
100e2xy100 − 0.01x2(a) Prove that the equation has at least one real solution.(b) Use your graphing device to find the solution correct to three decimal places.
ln x − 3 2 2x(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.
cos x − x3(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.
sin x − x2 2 x, s1, 2d Use the Intermediate Value Theorem to show that there is a solution of the given equation in the specified interval.
ex − 3 2 2x, s0, 1d Use the Intermediate Value Theorem to show that there is a solution of the given equation in the specified interval.
s3 x − 1 2 x, s0, 1d Use the Intermediate Value Theorem to show that there is a solution of the given equation in the specified interval.
x4 1 x 2 3 − 0, s1, 2d Use the Intermediate Value Theorem to show that there is a solution of the given equation in the specified interval.
Suppose f is continuous on f1, 5g and the only solutions of the equation f sxd − 6 are x − 1 and x − 4. If f s2d − 8, explain why f s3d . 6.
If f sxd − x2 1 10 sin x, show that there is a number c such that f scd − 1000.
Suppose that a function f is continuous on [0, 1] except at 0.25 and that f s0d − 1 and f s1d − 3. Let N − 2. Sketch two possible graphs off, one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f might still satisfy the conclusion of the
For what value of the constant c is the function f continuous on s2`, `d? Sex + 2x if x
The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet iswhere M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r? GMr if r
Find the numbers at which the functionis discontinuous. At which of these points is f continuous from the right, from the left, or neither? Sketch the graph of f . x+2 if x < 0 if 0 x 1 f(x)=e 2 x if x 1
Show that f is continuous on s2`, `d. f(x) = sin x if x
Show that f is continuous on s2`, `d. f(x) = 2 x if x < 1 x if x = 1
Use continuity to evaluate the limit.Drug resistance As we have previously noted (page 25), if p is the current frequency of the resistance gene in a model for the spread of drug resistance, then the frequency in the next generation is p2 2 2p p2 2 2 What is the limit of this function as p l 12 ?
Use continuity to evaluate the limit.lim x l 1 ex 22x
Use continuity to evaluate the limit.lim x l sinsx 1 sin xd
Locate the discontinuities of the function and illustrate by graphing.y − lnstan2xd 29. Use continuity to evaluate the limit. Xx lim
Locate the discontinuities of the function and illustrate by graphing. y= 1 1 + e/x
Explain, using Theorems 4, 5, 6, and 8, why the function is continuous at every number in its domain. State the domain.Fsxd − sinscosssin xdd
Explain, using Theorems 4, 5, 6, and 8, why the function is continuous at every number in its domain. State the domain.Gstd − lnst 4 2 1d
Explain, using Theorems 4, 5, 6, and 8, why the function is continuous at every number in its domain. State the domain.hsxd −sin x x 1 1
Explain, using Theorems 4, 5, 6, and 8, why the function is continuous at every number in its domain. State the domain.Lstd − e25t cos 2t
Explain, using Theorems 4, 5, 6, and 8, why the function is continuous at every number in its domain. State the domain.Gsxd − s3 x s1 1 x3d
Explain, using Theorems 4, 5, 6, and 8, why the function is continuous at every number in its domain. State the domain.Rsxd − x2 1 s2x 2 1
Explain why the function is discontinuous at the given numbera. Sketch the graph of the function. f(x)= = 2x2-5x-3 x-3 6 if x3 a = 3 if x = 3
Explain why the function is discontinuous at the given numbera. Sketch the graph of the function. COS X if x < 0 f(x)= = 0 if x = 0 a=0 1-x if x>0
Explain why the function is discontinuous at the given numbera. Sketch the graph of the function. f(x) = 1 2 x 1 if x 1 if x=1 a = 1
Explain why the function is discontinuous at the givennumbera. Sketch the graph of the function. if x
Use the definition of continuity and the properties of limits to show that the following function is continuous on the interval s2, `d.f sxd −2x 1 3 x 2 2
f sxd − sx 1 2x3 d4, a − 21 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.
f sxd − 3x4 2 5x 1 s3 x2 1 4 , a − 2 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.
If f and t are continuous functions with f s3d − 5 and limxl3 f2 f sxd 2 tsxdg − 4, find ts3d.
Explain why each function is continuous or discontinuous.(a) The temperature in New York City as a function of time(b) The population of New York City as a function of time(c) The temperature at a specific time as a function of the distance due west from New York City(d) The altitude above sea
A parking lot charges $3 for the first hour (or part of an hour) and $2 for each succeeding hour (or part), up to a daily maximum of $10.(a) Sketch a graph of the cost of parking at this lot as a function of the time parked there.(b) Discuss the discontinuities of this function and their
Squirrel population The graph of a population Pstd of squirrels is shown. Identify the discontinuities of P and comment on when and why they occur. P 30 25 20 0 2 3 3 4 5 t (days)
Drug concentration A patient is injected with a drug every 12 hours. The graph shows the concentration Cstd of the drug in the bloodstream after t hours.(a) At what values of t does C have discontinuities?(b) What type of discontinuity does C have? 88 80 C (mg/mL) 0 12 24 36 48 t (hours)
Neither left nor right continuous at 22, continuous only from the left at 2 Sketch the graph of a function f that is continuous except for the stated discontinuity.
Removable discontinuity at 3, jump discontinuity at 5 Sketch the graph of a function f that is continuous except for the stated discontinuity.
Discontinuities at 21 and 4, but continuous from the left at 21 and from the right at 4 Sketch the graph of a function f that is continuous except for the stated discontinuity.
Discontinuous, but continuous from the right, at 2 Sketch the graph of a function f that is continuous except for the stated discontinuity.
From the graph of t, state the intervals on which t is continuous. -4 2 4 6 8 x *
(a) From the graph of f , state the numbers at which f is discontinuous and explain why.(b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither. 4 2 x
If f is continuous on s2`, `d, what can you say about its graph?
Write an equation that expresses the fact that a function f is continuous at the number 4.
The figure shows a fixed circle C1 with equation sx 2 1d2 1 y2 − 1 and a shrinking circle C2 with radius r and center the origin. P is the point s0, rd, Q is the upper point of intersection of the two circles, and R is the point of intersection of the line PQ and the x-axis. What happens to R as
Is there a number a such that lim x l 22 3x2 1 ax 1 a 1 3 x2 1 x 2 2 exists? If so, find the value of a and the value of the limit.

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