A First Course in Differential Equations with Modeling Applications (MindTap Course List)
11th Edition
ISBN: 9781305965720
Author: Dennis G. Zill
Publisher: Cengage Learning
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Question
Chapter 1.1, Problem 60E
(a)
To determine
A constant solution of the given differential equation.
(b)
To determine
The intervals on the y-axis on which a nonconstant solution is increasing and decreasing.
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Exercises
Evaluate the following limits.
1. lim cot x/ln x
+01x
2. lim x² In x
+014
3. lim x*
x0+
4. lim (cos√√x)1/x
+014
5. lim x2/(1-cos x)
x10
6. lim e*/*
818
7. lim (secx - tan x)
x-x/2-
8. lim [1+(3/x)]*
x→∞0
In Exercises 1 through 3, let xo =
O and calculate P7(x) and R7(x).
1. f(x)=sin x, x in R.
2. f(x) = cos x, x in R.
3. f(x) = In(1+x), x≥0.
4. In Exercises 1, 2, and 3, for |x| 1, calculate a value of n such that P(x)
approximates f(x) to within 10-6.
5. Let (an)neN be a sequence of positive real numbers such that L =
lim (an+1/an) exists in R. If L < 1, show that an → 0. [Hint: Let
1111
L
iation
7. Let f be continuous on [a, b] and differentiable on (a, b). If lim f'(x)
xia
exists in R, show that f is differentiable at a and f'(a) = lim f'(x). A
similar result holds for b.
x-a
8. In reference to Corollary 5.4, give an example of a uniformly continuous
function on [0, 1] that is differentiable on (0, 1] but whose derivative is not
bounded there.
9. Recall that a fixed point of a function f is a point c such that f(c) = c.
(a) Show that if f is differentiable on R and f'(x)| x if x 1 and hence In(1+x) 0.
12. For 0 л/2. (Thus,
as x л/2 from the left, cos x is never large enough for x+cosx to be
greater than л/2 and cot x is never small enough for x + cot x to be less
than x/2.)
Chapter 1 Solutions
A First Course in Differential Equations with Modeling Applications (MindTap Course List)
Ch. 1.1 - In Problems 18 state the order of the given...Ch. 1.1 - In Problems 18 state the order of the given...Ch. 1.1 - In Problems 18 state the order of the given...Ch. 1.1 - In Problems 18 state the order of the given...Ch. 1.1 - In Problems 18 state the order of the given...Ch. 1.1 - In Problems 18 state the order of the given...Ch. 1.1 - In Problems 18 state the order of the given...Ch. 1.1 - In Problems 18 state the order of the given...Ch. 1.1 - In Problems 9 and 10 determine whether the given...Ch. 1.1 - In Problems 9 and 10 determine whether the given...
Ch. 1.1 - In Problems 1114 verify that the indicated...Ch. 1.1 - In Problems 1114 verify that the indicated...Ch. 1.1 - In Problems 1114 verify that the indicated...Ch. 1.1 - In Problems 1114 verify that the indicated...Ch. 1.1 - In Problems 1518 verify that the indicated...Ch. 1.1 - In Problems 1518 verify that the indicated...Ch. 1.1 - In Problems 1518 verify that the indicated...Ch. 1.1 - In Problems 1518 verify that the indicated...Ch. 1.1 - In Problems 19 and 20 verify that the indicated...Ch. 1.1 - In Problems 19 and 20 verify that the indicated...Ch. 1.1 - In Problems 2124 verify that the indicated family...Ch. 1.1 - In Problems 2124 verify that the indicated family...Ch. 1.1 - In Problems 2124 verify that the indicated family...Ch. 1.1 - In Problems 2124 verify that the indicated family...Ch. 1.1 - In Problems 2528 use (12) to verify that the...Ch. 1.1 - In Problems 2528 use (12) to verify that the...Ch. 1.1 - In Problems 2528 use (12) to verify that the...Ch. 1.1 - In Problems 2528 use (12) to verify that the...Ch. 1.1 - Verify that the piecewise-defined function...Ch. 1.1 - In Example 7 we saw that y=1(x)=25x2 and...Ch. 1.1 - In Problems 31-34 find values of m so that the...Ch. 1.1 - In Problems 31-34 find values of m so that the...Ch. 1.1 - In Problems 31-34 find values of m so that the...Ch. 1.1 - In Problems 31-34 find values of m so that the...Ch. 1.1 - In Problems 35 and 36 find values of m so that the...Ch. 1.1 - In Problems 35 and 36 find values of m so that the...Ch. 1.1 - In Problems 3740 use the concept that y = c, x ...Ch. 1.1 - In Problems 3740 use the concept that y = c, x ...Ch. 1.1 - In Problems 3740 use the concept that y = c, x ...Ch. 1.1 - In Problems 3740 use the concept that y = c, x ...Ch. 1.1 - Prob. 41ECh. 1.1 - In Problems 41 and 42 verify that the indicated...Ch. 1.1 - Prob. 43ECh. 1.1 - Make up a differential equation that you feel...Ch. 1.1 - What function do you know from calculus is such...Ch. 1.1 - What function (or functions) do you know from...Ch. 1.1 - The function y = sin x is an explicit solution of...Ch. 1.1 - Discuss why it makes intuitive sense to presume...Ch. 1.1 - In Problems 49 and 50 the given figure represents...Ch. 1.1 - In Problems 49 and 50 the given figure represents...Ch. 1.1 - The graphs of members of the one-parameter family...Ch. 1.1 - Prob. 52ECh. 1.1 - In Example 7 the largest interval I over which the...Ch. 1.1 - In Problem 21 a one-parameter family of solutions...Ch. 1.1 - Discuss, and illustrate with examples, how to...Ch. 1.1 - The differential equation x(y)2 4y 12x3 = 0 has...Ch. 1.1 - Prob. 57ECh. 1.1 - Find a linear second-order differential equation...Ch. 1.1 - Prob. 59ECh. 1.1 - Prob. 60ECh. 1.1 - Consider the differential equation dy/dx = y(a ...Ch. 1.1 - Consider the differential equation y = y2 + 4. (a)...Ch. 1.2 - In Problems 1 and 2, y = 1/(1 + c1ex) is a...Ch. 1.2 - In Problems 1 and 2, y = 1/(1 + c1ex) is a...Ch. 1.2 - In Problems 36, y = 1/(x2 + c) is a one-parameter...Ch. 1.2 - In Problems 36, y = 1/(x2 + c) is a one-parameter...Ch. 1.2 - In Problems 36, y = 1/(x2 + c) is a one-parameter...Ch. 1.2 - In Problems 36, y = 1/(x2 + c) is a one-parameter...Ch. 1.2 - In Problems 710, x = c1 cos t + c2 sin t is a...Ch. 1.2 - Prob. 8ECh. 1.2 - In Problems 710, x = c1 cos t + c2 sin t is a...Ch. 1.2 - In Problems 710, x = c1 cos t + c2 sin t is a...Ch. 1.2 - In Problems 1114, y = c1ex + c2ex is a...Ch. 1.2 - In Problems 1114, y = c1ex + c2ex is a...Ch. 1.2 - In Problems 1114, y = c1ex + c2ex is a...Ch. 1.2 - In Problems 1114, y = c1ex + c2ex is a...Ch. 1.2 - In Problems 15 and 16 determine by inspection at...Ch. 1.2 - In Problems 15 and 16 determine by inspection at...Ch. 1.2 - In Problems 1724 determine a region of the...Ch. 1.2 - In Problems 1724 determine a region of the...Ch. 1.2 - In Problems 1724 determine a region of the...Ch. 1.2 - In Problems 1724 determine a region of the...Ch. 1.2 - In Problems 1724 determine a region of the...Ch. 1.2 - In Problems 1724 determine a region of the...Ch. 1.2 - In Problems 1724 determine a region of the...Ch. 1.2 - In Problems 1724 determine a region of the...Ch. 1.2 - In Problems 2528 determine whether Theorem 1.2.1...Ch. 1.2 - In Problems 2528 determine whether Theorem 1.2.1...Ch. 1.2 - In Problems 2528 determine whether Theorem 1.2.1...Ch. 1.2 - In Problems 2528 determine whether Theorem 1.2.1...Ch. 1.2 - (a) By inspection find a one-parameter family of...Ch. 1.2 - (a) Verify that y = tan (x + c) is a one-parameter...Ch. 1.2 - (a) Verify that y = 1 /(x + c) is a one-parameter...Ch. 1.2 - (a) Show that a solution from the family in part...Ch. 1.2 - (a) Verify that 3x2 y2 = c is a one-parameter...Ch. 1.2 - (a) Use the family of solutions in part (a) of...Ch. 1.2 - In Problems 3538 the graph of a member of a family...Ch. 1.2 - In Problems 3538 the graph of a member of a family...Ch. 1.2 - In Problems 3538 the graph of a member of a family...Ch. 1.2 - In Problems 3538 the graph of a member of a family...Ch. 1.2 - In Problems 3944, y = c1 cos 2x + c2 sin 2x is a...Ch. 1.2 - In Problems 3944, y = c1 cos 2x + c2 sin 2x is a...Ch. 1.2 - In Problems 3944, y = c1 cos 2x + c2 sin 2x is a...Ch. 1.2 - In Problems 3944, y = c1 cos 2x + c2 sin 2x is a...Ch. 1.2 - In Problems 3944, y = c1 cos 2x + c2 sin 2x is a...Ch. 1.2 - In Problems 3944, y = c1 cos 2x + c2 sin 2x is a...Ch. 1.2 - Find a function whose graph at each point (x, y)...Ch. 1.2 - Prob. 46ECh. 1.2 - Consider the initial-value problem y = x 2y, y(0)...Ch. 1.2 - Show that x=0y1t3+1dt is an implicit solution of...Ch. 1.2 - Prob. 49ECh. 1.2 - Suppose that the first-order differential equation...Ch. 1.2 - The functions y(x)=116x4, x and...Ch. 1.3 - Under the same assumptions that underlie the model...Ch. 1.3 - The population model given in (1) fails to take...Ch. 1.3 - Using the concept of net rate introduced in...Ch. 1.3 - Modify the model in Problem 3 for net rate at...Ch. 1.3 - A cup of coffee cools according to Newtons law of...Ch. 1.3 - The ambient temperature Tm in (3) could be a...Ch. 1.3 - Suppose a student carrying a flu virus returns to...Ch. 1.3 - At a time denoted as t = 0 a technological...Ch. 1.3 - Suppose that a large mixing tank initially holds...Ch. 1.3 - Suppose that a large mixing tank initially holds...Ch. 1.3 - What is the differential equation in Problem 10,...Ch. 1.3 - Generalize the model given in equation (8) of this...Ch. 1.3 - Suppose water is leaking from a tank through a...Ch. 1.3 - The right-circular conical tank shown in Figure...Ch. 1.3 - A series circuit contains a resistor and an...Ch. 1.3 - A series circuit contains a resistor and a...Ch. 1.3 - For high-speed motion through the airsuch as the...Ch. 1.3 - A cylindrical barrel s feet in diameter of weight...Ch. 1.3 - After a mass m is attached to a spring, it...Ch. 1.3 - In Problem 19, what is a differential equation for...Ch. 1.3 - A small single-stage rocket is launched vertically...Ch. 1.3 - In Problem 21, the mass m(t) is the sum of three...Ch. 1.3 - By Newtons universal law of gravitation the...Ch. 1.3 - Suppose a hole is drilled through the center of...Ch. 1.3 - Learning Theory In the theory of learning, the...Ch. 1.3 - Forgetfulness In Problem 25 assume that the rate...Ch. 1.3 - Infusion of a Drug A drug is infused into a...Ch. 1.3 - Tractrix A motorboat starts at the origin and...Ch. 1.3 - Reflecting surface Assume that when the plane...Ch. 1.3 - Reread Problem 45 in Exercises 1.1 and then give...Ch. 1.3 - Prob. 31ECh. 1.3 - Prob. 32ECh. 1.3 - Prob. 33ECh. 1.3 - Rotating Fluid As shown in Figure 1.3.24(a), a...Ch. 1.3 - Prob. 35ECh. 1.3 - Prob. 36ECh. 1.3 - Let It snow The snowplow problem is a classic and...Ch. 1.3 - Population Dynamics Suppose that dP/dt = 0.15 P(t)...Ch. 1.3 - Radioactive Decay Suppose that dA/dt = 0.0004332...Ch. 1.3 - Reread this section and classify each mathematical...Ch. 1 - In Problems 1 and 2 fill in the blank and then...Ch. 1 - In Problems 1 and 2 fill in the blank and then...Ch. 1 - In Problems 3 and 4 fill in the blank and then...Ch. 1 - In Problems 3 and 4 fill in the blank and then...Ch. 1 - In Problems 5 and 6 compute y and y and then...Ch. 1 - In Problems 5 and 6 compute y and y and then...Ch. 1 - In Problems 712 match each of the given...Ch. 1 - In Problems 712 match each of the given...Ch. 1 - In Problems 712 match each of the given...Ch. 1 - In Problems 712 match each of the given...Ch. 1 - In Problems 712 match each of the given...Ch. 1 - In Problems 712 match each of the given...Ch. 1 - In Problems 13 and 14 determine by inspection at...Ch. 1 - In Problems 13 and 14 determine by inspection at...Ch. 1 - In Problems 15 and 16 interpret each statement as...Ch. 1 - In Problems 15 and 16 interpret each statement as...Ch. 1 - (a) Give the domain of the function y = x2/3. (b)...Ch. 1 - (a) Verify that the one-parameter family y2 2y =...Ch. 1 - The function y = x 2/x is a solution of the DE xy...Ch. 1 - Suppose that y(x) denotes a solution of the...Ch. 1 - A differential equation may possess more than one...Ch. 1 - What is the slope of the tangent line to the graph...Ch. 1 - In Problems 2326 verify that the indicated...Ch. 1 - In Problems 2326 verify that the indicated...Ch. 1 - In Problems 2326 verify that the indicated...Ch. 1 - In Problems 2326 verify that the indicated...Ch. 1 - In Problems 2730 use (12) of Section 1.1 to verify...Ch. 1 - In Problems 2730 use (12) of Section 1.1 to verify...Ch. 1 - In Problems 2730 use (12) of Section 1.1 to verify...Ch. 1 - In Problems 2730 use (12) of Section 1.1 to verify...Ch. 1 - In Problems 3134 verify that the indicated...Ch. 1 - In Problems 3134 verify that the indicated...Ch. 1 - In Problems 3134 verify that the indicated...Ch. 1 - In Problems 3134 verify that the indicated...Ch. 1 - In Problems 3538, y = c1e3x + c2ex 2x is a...Ch. 1 - In Problems 3538, y = c1e3x + c2ex 2x is a...Ch. 1 - In Problems 3538, y = c1e3x + c2ex 2x is a...Ch. 1 - In Problems 3538, y = c1e3x + c2ex 2x is a...Ch. 1 - The graph of a solution of a second-order...Ch. 1 - A tank in the form of a right-circular cylinder of...
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