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Chapter 9: Problem 11

Compare the values of \(d y\) and \(\Delta y\). \(y=0.5 x^{3} \quad x=2 \quad \Delta x=d x=0.1\)

Short Answer

Expert verified
The exact change in the function \(dy\) is 0.6 and the approximated change in the function \(\Delta y\) is 0.6305.

Step by step solution

01

Differentiation

The first step involves differentiation of the given function \(y\) with respect to \(x\). Thus, this gives: \(y' = dy/dx = 0.5 \times 3x^2 = 1.5x^2\).
02

Calculate the value of \(dy\)

Using the value \(x=2\) and \(dx=0.1\), the exact change in \(y\) i.e. \(dy\) can be calculated using the formula: \(dy = y'(2) \times dx = 1.5 \times 2^2 \times 0.1 = 0.6\). Thus, \(dy\) equals 0.6.
03

Calculate the value of \(\Delta y\)

Now we will calculate \(\Delta y\), the approximate change in \(y\), which is given by: \(\Delta y = (y(x + dx) - y(x))\). Substituting the given values: \(\Delta y = y((2+0.1))-y(2) = 0.5 \times (2.1)^3 - 0.5 \times 2^3 = 0.5 \times 9.261-0.5 \times 8 = 0.6305 - 4 = 0.6305\). Thus, \(\Delta y\) equals 0.6305.

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Most popular questions from this chapter

The cost function for a certain model of personal digital assistant (PDA) is given by \(C=13.50 x+45,750\), where \(C\) is measured in dollars and \(x\) is the number of PDAs produced. (a) Find the average cost function \(\bar{C}\). (b) Find \(\bar{C}\) when \(x=100\) and \(x=1000\). (c) Determine the limit of the average cost function as \(x\) approaches infinity. Interpret the limit in the context of the problem.

Let \(x=2\) and complete the table for the function. $$ \begin{array}{|c|c|c|c|c|} \hline d x=\Delta x & d y & \Delta y & \Delta y-d y & d y / \Delta y \\ \hline 1.000 & & & & \\ \hline 0.500 & & & & \\ \hline 0.100 & & & & \\ \hline 0.010 & & & & \\ \hline 0.001 & & & & \\ \hline \end{array} $$ \(y=\frac{1}{x}\)

Marginal Analysis, use differentials to approximate the change in cost, revenue, or profit corresponding to an increase in sales of one unit. For instance, in Exercise 29, approximate the change in cost as \(x\) increases from 12 units to 13 units. Then use a graphing utility to graph the function, and use the trace feature to verify your result. \(C=0.05 x^{2}+4 x+10 \quad x=12\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=\frac{x^{2}}{x^{2}+3}\)

Find the differential \(d y\). \(y=(4 x-1)^{3}\)
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