University Calculus: Early Transcendentals, Books a la Carte Edition (3rd Edition)
3rd Edition
ISBN: 9780321999610
Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.
Publisher: PEARSON
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Chapter 12, Problem 3AAE
To determine
Verify that
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2) Compute the following anti-derivative.
√1x4 dx
Question 3 (5pt): A chemical reaction. In an elementary chemical reaction,
single molecules of two reactants A and B form a molecule of the product C :
ABC. The law of mass action states that the rate of reaction is proportional
to the product of the concentrations of A and B:
d[C]
dt
= k[A][B]
(where k is a constant positive number). Thus, if the initial concentrations are
[A] =
= a moles/L and [B] = b moles/L we write x = [C], then we have
(E):
dx
dt
=
k(ax)(b-x)
1
(a) Write the differential equation (E) with separate variables, i.e. of the form
f(x)dx = g(t)dt.
(b) Assume first that a b. Show that
1
1
1
1
=
(a - x) (b - x)
-
a) a - x
b - x
b)
(c) Find an antiderivative for the function f(x) = (a-x) (b-x) using the previous
question.
(d) Solve the differentiel equation (E), i.e. find x as a function of t. Use the fact
that the initial concentration of C is 0.
(e) Now assume that a = b. Find x(t) assuming that a = b. How does this
expression for x(t) simplify if it is known that [C] =…
3) Find the volume of the solid that lies inside both the sphere x² + y² + z²
cylinder x²+y² = 1.
= 4 and the
Chapter 12 Solutions
University Calculus: Early Transcendentals, Books a la Carte Edition (3rd Edition)
Ch. 12.1 - Motion in the Plane In Exercises 58, r(t) is the...Ch. 12.1 - Motion in the Plane
In Exercises 5–8, r(t) is the...Ch. 12.1 - In Exercises 58, r(t) is the position of a...Ch. 12.1 - In Exercises 5–8, r(t) is the position of a...Ch. 12.1 - Prob. 5ECh. 12.1 - Prob. 6ECh. 12.1 - Exercises 9–12 give the position vectors of...Ch. 12.1 - Prob. 8ECh. 12.1 - In Exercises 13–18, r(t) is the position of a...Ch. 12.1 - Prob. 10E
Ch. 12.1 - In Exercises 13–18, r(t) is the position of a...Ch. 12.1 - Prob. 12ECh. 12.1 - Prob. 13ECh. 12.1 - In Exercises 13–18, r(t) is the position of a...Ch. 12.1 - In Exercises 1922, r(t) is the position of a...Ch. 12.1 - In Exercises 19–22, r(t) is the position of a...Ch. 12.1 - In Exercises 19–22, r(t) is the position of a...Ch. 12.1 - Prob. 18ECh. 12.1 - As mentioned in the text, the tangent line to a...Ch. 12.1 - Prob. 20ECh. 12.1 - Tangents to Curves
As mentioned in the text, the...Ch. 12.1 - Prob. 22ECh. 12.1 - Motion along a circle Each of the following...Ch. 12.1 - Motion along a circle Show that the vector-valued...Ch. 12.1 - Prob. 25ECh. 12.1 - Motion along a cycloid A particle moves in the...Ch. 12.1 - Prob. 27ECh. 12.1 - Prob. 28ECh. 12.1 - Prob. 29ECh. 12.1 - Prob. 30ECh. 12.1 - Component test for continuity at a point Show that...Ch. 12.1 - Limits of cross products of vector functions...Ch. 12.1 - Differentiable vector functions are continuous...Ch. 12.1 - Constant Function Rule Prove that if u is the...Ch. 12.2 - Evaluate the integrals in Exercises 1–10.
1.
Ch. 12.2 - Evaluate the integrals in Exercises 1–10.
2.
Ch. 12.2 - Evaluate the integrals in Exercises 1–10.
3.
Ch. 12.2 - Evaluate the integrals in Exercises 1–10.
4.
Ch. 12.2 - Evaluate the integrals in Exercises 1–10.
5.
Ch. 12.2 - Evaluate the integrals in Exercises 1–10.
6.
Ch. 12.2 - Evaluate the integrals in Exercises 110. 7....Ch. 12.2 - Evaluate the integrals in Exercises 1–10.
8.
Ch. 12.2 - Prob. 9ECh. 12.2 - Prob. 10ECh. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Prob. 16ECh. 12.2 - At time t = 0, a particle is located at the point...Ch. 12.2 - Prob. 18ECh. 12.2 - Prob. 19ECh. 12.2 - Range and height versus speed
Show that doubling a...Ch. 12.2 - Flight time and height A projectile is fired with...Ch. 12.2 - Prob. 22ECh. 12.2 - Prob. 23ECh. 12.2 - Beaming electrons An electron in a TV tube is...Ch. 12.2 - Prob. 25ECh. 12.2 - Finding muzzle speed Find the muzzle speed of a...Ch. 12.2 - Prob. 27ECh. 12.2 - Colliding marbles The accompanying figure shows an...Ch. 12.2 - Firing from (x0, y0) Derive the equations
(see...Ch. 12.2 - Where trajectories crest For a projectile fired...Ch. 12.2 - Prob. 31ECh. 12.2 - Prob. 32ECh. 12.2 - Prob. 33ECh. 12.2 - Products of scalar and vector functions Suppose...Ch. 12.2 - Prob. 35ECh. 12.2 - The Fundamental Theorem of Calculus The...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - Prob. 7ECh. 12.3 - In Exercises 1–8, find the curve’s unit tangent...Ch. 12.3 - Find the point on the curve
at a distance 26...Ch. 12.3 - Find the point on the curve
at a distance 13...Ch. 12.3 - In Exercises 11–14, find the arc length parameter...Ch. 12.3 - In Exercises 11–14, find the arc length parameter...Ch. 12.3 - In Exercises 11–14, find the arc length parameter...Ch. 12.3 - In Exercises 11–14, find the arc length parameter...Ch. 12.3 - Arc length Find the length of the curve
from (0,...Ch. 12.3 - Length of helix The length of the turn of the...Ch. 12.3 - Length is independent of parametrization To...Ch. 12.3 - The involute of a circle If a siring wound around...Ch. 12.3 - Prob. 20ECh. 12.3 - Distance along a line Show that if u is a unit...Ch. 12.3 - Prob. 22ECh. 12.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 12.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 12.4 - Find T, N, and for the plane curves in Exercises...Ch. 12.4 - Find T, N, and κ for the plane curves in Exercises...Ch. 12.4 - Prob. 5ECh. 12.4 - Prob. 6ECh. 12.4 - Prob. 7ECh. 12.4 - Prob. 8ECh. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Prob. 10ECh. 12.4 - Prob. 11ECh. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Find T, N, and κ for the space curves in Exercises...Ch. 12.4 - Prob. 16ECh. 12.4 - Show that the parabola , has its largest curvature...Ch. 12.4 - Show that the ellipse x = a cos t, y = b sin t, a...Ch. 12.4 - Prob. 19ECh. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - Prob. 25ECh. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Prob. 28ECh. 12.5 - In Exercises 1 and 2, write a in the form a = aTT...Ch. 12.5 - In Exercises 1 and 2, write a in the form a = aTT...Ch. 12.5 - In Exercises 36, write a in the form a = aTT + aNN...Ch. 12.5 - Prob. 4ECh. 12.5 - In Exercises 3–6, write a in the form a = aTT +...Ch. 12.5 - In Exercises 3–6, write a in the form a = aTT +...Ch. 12.5 - In Exercises 7 and 8, find r, T, N, and B at the...Ch. 12.5 - Prob. 8ECh. 12.5 - The speedometer on your car reads a steady 35 mph....Ch. 12.5 - Prob. 10ECh. 12.5 - Can anything be said about the speed of a particle...Ch. 12.5 - An object of mass m travels along the parabola y =...Ch. 12.5 - Prob. 13ECh. 12.5 - Prob. 14ECh. 12.5 - Prob. 15ECh. 12.5 - Prob. 16ECh. 12.6 - Prob. 1ECh. 12.6 - Prob. 2ECh. 12.6 - Prob. 3ECh. 12.6 - Prob. 4ECh. 12.6 - Prob. 5ECh. 12.6 - Prob. 6ECh. 12.6 - Prob. 7ECh. 12.6 - Prob. 8ECh. 12.6 - Prob. 9ECh. 12.6 - Prob. 10ECh. 12.6 - Prob. 11ECh. 12.6 - Prob. 12ECh. 12.6 - Prob. 13ECh. 12.6 - Prob. 14ECh. 12.6 - Prob. 15ECh. 12.6 - Prob. 16ECh. 12 - Prob. 1GYRCh. 12 - Prob. 2GYRCh. 12 - Prob. 3GYRCh. 12 - Prob. 4GYRCh. 12 - Prob. 5GYRCh. 12 - Prob. 6GYRCh. 12 - Prob. 7GYRCh. 12 - Prob. 8GYRCh. 12 - Prob. 9GYRCh. 12 - Prob. 10GYRCh. 12 - Prob. 11GYRCh. 12 - Prob. 12GYRCh. 12 - Prob. 13GYRCh. 12 - In Exercises 1 and 2, graph the curves and sketch...Ch. 12 - Prob. 2PECh. 12 - Prob. 3PECh. 12 - Prob. 4PECh. 12 - Prob. 5PECh. 12 - Prob. 6PECh. 12 - Prob. 7PECh. 12 - Prob. 8PECh. 12 - Prob. 9PECh. 12 - Prob. 10PECh. 12 - Prob. 11PECh. 12 - Prob. 12PECh. 12 - Prob. 13PECh. 12 - Prob. 14PECh. 12 - Prob. 15PECh. 12 - Prob. 16PECh. 12 - Prob. 17PECh. 12 - Prob. 18PECh. 12 - Prob. 19PECh. 12 - In Exercises 17-20, find T, N, B, and k at the...Ch. 12 - Prob. 21PECh. 12 - Prob. 22PECh. 12 - Prob. 23PECh. 12 - Prob. 24PECh. 12 - Prob. 25PECh. 12 - Find equations for the osculating, normal, and...Ch. 12 - Find parametric equations for the line that is...Ch. 12 - Prob. 28PECh. 12 - Prob. 29PECh. 12 - Prob. 30PECh. 12 - Prob. 1AAECh. 12 - Suppose the curve in Exercise 1 is replaced by the...Ch. 12 - Prob. 3AAECh. 12 - Prob. 4AAECh. 12 - Prob. 5AAECh. 12 - Prob. 6AAECh. 12 - Prob. 7AAECh. 12 - Prob. 8AAE
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