Essential Calculus: Early Transcendentals
2nd Edition
ISBN: 9781133112280
Author: James Stewart
Publisher: Cengage Learning
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Chapter T, Problem 7CDT
(a)
To determine
The value of the function
(b)
To determine
The value of the function
(c)
To determine
The value of the function
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The parametric equations of the function are given asx=asin²0, y = acos). Calculate
[Let: a=anumerical coefficient]
dy
d²y
and
dx
dx2
A tank contains 200 gal of fresh water. A solution containing 4 lb/gal of soluble
lawn fertilizer runs into the tank at the rate of 1 gal/min, and the mixture is
pumped out of the tank at the rate of 5 gal/min. Find the maximum amount of
fertilizer in the tank and the time required to reach the maximum.
Find the time required to reach the maximum amount of fertilizer in the tank.
t=
min
(Type an integer or decimal rounded to the nearest tenth as needed.)
Chapter T Solutions
Essential Calculus: Early Transcendentals
Ch. T - Evaluate each expression without using a...Ch. T - Simplify each expression. Write your answer...Ch. T - Expand and simplify.
3(x + 6) + 4(2x − 5)
(x +...Ch. T - Factor each expression.
4x2 − 25
2x2 + 5x − 12
x3...Ch. T - Simplify the rational expression.
Ch. T - Rationalize the expression and simplify.
(a)
(b)...Ch. T - Rewrite by completing the square.
x2 + x + 1
2x2...Ch. T - Solve the equation. (Find only the real...Ch. T - Solve each inequality. Write your answer using...Ch. T - State whether each equation is true or...
Ch. T - Find an equation for the line that passes through...Ch. T - Prob. 2BDTCh. T - Prob. 3BDTCh. T - Let A(−7, 4) and B(5, −12) be points in the...Ch. T - Sketch the region in the xy-plane defined by the...Ch. T - The graph of a function f is given at the left....Ch. T - If f(x) = x3, evaluate the difference quotient...Ch. T - Find the domain of the function. (a)...Ch. T - How are graphs of the functions obtained from the...Ch. T - Without using a calculator, make a rough sketch of...Ch. T - Let
Evaluate f(−2) and f(1).
Sketch the graph of...Ch. T - Prob. 7CDTCh. T - Prob. 1DDTCh. T - Convert from radians to degrees. (a) 5/6 (b) 2Ch. T - Find the length of an arc of a circle with radius...Ch. T - Find the exact values. (a) tan(/3) (b) sin(7/6)...Ch. T - Express the lengths a and b in the figure in terms...Ch. T - If sinx=13 and secy=54, where x and y lie between...Ch. T - Prove the identities.
tan θ sin θ + cos θ = sec...Ch. T - Find all values of x such that sin 2x = sin x and...Ch. T - Sketch the graph of the function y = 1 + sin 2x...
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- Thumbi Irrigation Scheme in Mzimba district is under threat of flooding. In order to mitigate against the problem, authorities have decided to construct a flood protection bund (Dyke). Figure 1 is a cross section of a 300m long proposed dyke; together with its foundation (key). Survey data for the proposed site of the dyke are presented in Table 1. Table 2 provides swelling and shrinkage factors for the fill material that has been proposed. The dyke dimensions that are given are for a compacted fill. (1) Assume you are in the design office, use both the Simpson Rule and Trapezoidal Rule to compute the total volume of earthworks required. (Assume both the dyke and the key will use the same material). (2) If you are a Contractor, how many days will it take to finish hauling the computed earthworks using 3 tippers of 12m³ each? Make appropriate assumptions. DIKE CROSS SECTION OGL KEY (FOUNDATION) 2m 1m 2m 8m Figure 1: Cross section of Dyke and its foundation 1.5m from highest OGL 0.5m…arrow_forwardThe parametric equations of the function are given as x = 3cos 0 - sin³0 and y = 3sin 0 - cos³0. dy d2y Calculate and dx dx².arrow_forward(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z ≤ 3}. Calculate the integral f(x, y, z) dv. Earrow_forward
- (12 points) Let E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}. (a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such that (x, y, z) (psin cos 0, psin sin 0, p cos) € E. (b) (8 points) Calculate the integral E xyz dV using spherical coordinates.arrow_forward(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}. Calculate the integral y, f(x, y, z) dV.arrow_forward(14 points) Let f: R3 R and T: R3. →R³ be defined by f(x, y, z) = ln(x²+ y²+2²), T(p, 0,4)=(psin cos 0, psin sin, pcos). (a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the gradient Vg directly, i.e. without using the chain rule. (b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4). (c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate Vg(r,0,4).arrow_forward
- (10 points) Let S be the upper hemisphere of the unit sphere x² + y²+2² = 1. Let F(x, y, z) = (x, y, z). Calculate the surface integral J F F-dS. Sarrow_forward(8 points) Calculate the following line integrals. (a) (4 points) F Fds where F(x, y, z) = (x, y, xy) and c(t) = (cost, sint, t), tЄ [0,π] . (b) (4 points) F. Fds where F(x, y, z) = (√xy, e³, xz) where c(t) = (t², t², t), t = [0, 1] .arrow_forwardreview help please and thank you!arrow_forward
- (10 points) Let S be the surface that is part of the sphere x² + y²+z² = 4 lying below the plane 2√3 and above the plane z-v -√3. Calculate the surface area of S.arrow_forward(8 points) Let D = {(x, y) | 0 ≤ x² + y² ≤4}. Calculate == (x² + y²)³/2dA by making a change of variables to polar coordinates, i.e. x=rcos 0, y = r sin 0.arrow_forwardx² - y² (10 points) Let f(x,y): = (a) (6 points) For each vector u = (1, 2), calculate the directional derivative Duƒ(1,1). (b) (4 points) Determine all unit vectors u for which Duf(1, 1) = 0.arrow_forward
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