CALCULUS WITH APPLICATIONS
11th Edition
ISBN: 2818440028625
Author: Lial
Publisher: ELSEVIER
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Chapter T, Problem 26DT
To determine
To simplify: The expression
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Find the absolute maximum and minimum of f(x, y) = x + y within the domain x² + y² ≤ 4.
Please show your answers to at least 4 decimal places. Enter DNE if the value does not exist.
1. Absolute minimum of f(x, y) is
Suppose that one factory inputs its goods from two different plants, A and B, with different costs, 3 and 7
each respective. And suppose the price function in the market is decided as p(x, y) = 100 - x - y where
I and y are the demand functions and 0 < x,y. Then as
x =
y =
the factory can attain the maximum profit,
Evaluate the following integrals, showing all your working
Chapter T Solutions
CALCULUS WITH APPLICATIONS
Ch. T - Prob. 1DTCh. T - Simplify .
Ch. T - Let x be the number of apples and y be the number...Ch. T - Let s be the number of students and p be the...Ch. T - Solve for k: 7k + 8 = –4(3 – k).
Ch. T - Solve for
Ch. T - Write in interval notation: -2 < x ≤ 5.
Ch. T - Using the variable x, write the following interval...Ch. T - Solve for y: 5(y – 2) + 1 ≤ 7y + 8.
Ch. T - Solve for .
Ch. T - Carry out the operations and simplify: (5y2 - 6y –...Ch. T - Multiply out and simplify (x2 – 2x + 3)(x + 1).
Ch. T - Multiply out and simplify (a – 2b)2.
Ch. T - Factor 3pq + 6p2q + 9pq2.
Ch. T - Prob. 15DTCh. T - Perform the operation and simplify: .
Ch. T - Perform the operation and simplify:.
Ch. T - Solve for x: 3x2 + 4x = 1.
Ch. T - Solve for z: .
Ch. T - Simplify
Ch. T - Simplify
Ch. T - Simplify as a single term without negative...Ch. T - Prob. 23DTCh. T - Simplify
Ch. T - Rationalize the denominator: .
Ch. T - Simplify .
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- Consider the function f(x) = 2x³-4x2-x+1. (a) Without doing a sketch, show that the cubic equation has at least one solution on the interval [0,1]. Use a theorem discussed in lectures, or see Section 1.8 of Calculus (7th ed) by Stewart. Ensure that the conditions of the theorem are satisfied (include this in your solution) (b) Now, by sketching the cubic (by hand or by computer), you should see that there is, in fact, exactly one zero in the interval [0,1]. Use Newton's method to find this zero accurate to 3 decimal places. You should include a sketch of the cubic, Newton's iteration formula, and the list of iterates. [Use a computer if possible, e.g., a spreadsheet or MatLab.]arrow_forwardEvaluate the following integrals, showing all your workingarrow_forwardDifferentiate the following functionarrow_forward
- Differentiate the following functionarrow_forwardA box with a square base and open top must have a volume of 13,500 cm³. Find the dimensions that minimise the amount of material used. Ensure you show your working to demonstrate that it is a minimum.arrow_forwardConsider the equation, f(x) = x*. (a) Using the trapezoidal method with 3 columns, estimate the value of the integral f² f(x)dx. (b) Using the trapezoidal method with 10 columns, estimate the value of the integral f² f(x)dx. You many need software to help you do this (e.g. MATLAB, Excel, Google sheets). (c) Use software to accurately calculate the integral (e.g. Wolfram alpha, Matlab). Using this answer, comment on the answers you found in parts a) and b).arrow_forward
- Question 1. (10 points) A researcher is studying tumours in mice. The growth rate for the volume of the tumour V(t) in cm³ is given by dV = 1.45V(2 In(V+1)). dt (a) (4 pts) Find all the equilibria and determine their stability using the stability condition. (b) (2 pts) Draw the phase plot f(V) versus V where f(V) = V'. You may find it helpful to use Desmos or Wolfram Alpha to plot the graph of f(V) versus V (both are free to use online), or you can plot it by hand if you like. On the plot identify each equilibrium as stable or unstable. (c) (4 pts) Draw direction arrows for the case where the tumour starts at size 3cm³ and for the case where the tumour starts at size 9cm³. Explain in biological terms what happens to the size of each of these tumours at time progresses.arrow_forwardFor the system consisting of the two planes:plane 1: -x + y + z = 0plane 2: 3x + y + 3z = 0a) Are the planes parallel and/or coincident? Justify your answer. What does this tell you about the solution to the system?b) Solve the system (if possible). Show a complete solution. If there is a line of intersection express it in parametric form.arrow_forwardQuestion 2: (10 points) Evaluate the definite integral. Use the following form of the definition of the integral to evaluate the integral: Theorem: Iff is integrable on [a, b], then where Ax = (ba)/n and x₂ = a + i^x. You might need the following formulas. IM³ L² (3x² (3x²+2x- 2x - 1)dx. n [f(z)dz lim f(x)Az a n→∞ i=1 n(n + 1) 2 n i=1 n(n+1)(2n+1) 6arrow_forward
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