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Chapter 3: Problem 71

Show that for any constants \(A\) and \(k\), the function \(y=A e^{k t}\) satisfies the equation \(d y / d t=k y\).

Short Answer

Expert verified
Yes, the function \(y = A e^{k t}\) satisfies \(\frac{d y}{d t} = k y\).

Step by step solution

01

Identify the Given Function and Problem

We are given the function \(y = A e^{k t}\) and need to show it satisfies the equation \(\frac{d y}{d t} = k y\). This means we need to compute the derivative of \(y\) with respect to \(t\) and show that it equals the product of \(k\) and \(y\).
02

Differentiate the Function with Respect to t

Calculate the derivative of the given function \(y = A e^{k t}\) with respect to \(t\). The derivative of \(e^{k t}\) is \(k e^{k t}\). Thus, the derivative of \(y\) is: \[\frac{d y}{d t} = A k e^{k t}\]
03

Substitute the Original Function

Substitute \(y = A e^{k t}\) back into the expression \(A k e^{k t}\) from the previous step: \[\frac{d y}{d t} = A k e^{k t} = k (A e^{k t})\]Since \(y = A e^{k t}\), we can see that \(k (A e^{k t}) = k y\).
04

Conclusion

From the steps above, we have shown that \(\frac{d y}{d t} = k y\) because the derivative \(A k e^{k t}\) can be expressed as \(k (A e^{k t}) = k y\). Therefore, \(y = A e^{k t}\) satisfies the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function
Exponential functions are a type of mathematical function where a constant base is raised to the power of a variable exponent. The base of natural exponential functions is the mathematical constant \(e\), which is approximately equal to 2.71828. These functions are written in the form \(y = A e^{k t}\), where \(A\) and \(k\) are constants, \(e\) is the base, and \(t\) is the variable representing time.
The exponential function shows how quantities grow or decay at a constant rate. They are commonly used to model natural phenomena such as population growth, radioactive decay, and interest compounding.
  • The base \(e\) is known for its amazing properties, especially affecting rates of change calculated as derivatives.
  • In the expression \(e^{k t}\), \(k\) determines the growth (\

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Most popular questions from this chapter

One side of a right triangle is known to be \(25 \mathrm{~cm}\) exactly. The angle opposite to this side is measured to be \(60^{\circ}\), with a possible error of \(\pm 0.5^{\circ}\). (a) Use differentials to estimate the errors in the adjacent side and the hypotenuse. (b) Estimate the percentage errors in the adjacent side and hypotenuse.

A metal rod \(15 \mathrm{~cm}\) long and \(5 \mathrm{~cm}\) in diameter is to be covered (except for the ends) with insulation that is \(0.1 \mathrm{~cm}\) thick. Use differentials to estimate the volume of insulation. [Hint: Let \(\Delta V\) be the change in volume of the rod.]

If the temperature \(T\) of a metal rod of length \(L\) is changed by an amount \(\Delta T\), then the length will change by the amount \(\Delta L=\alpha L \Delta T\), where \(\alpha\) is called the coefficient of linear expansion. For moderate changes in temperature \(\alpha\) is taken as constant. (a) Suppose that a rod \(40 \mathrm{~cm}\) long at \(20^{\circ} \mathrm{C}\) is found to be \(40.006 \mathrm{~cm}\) long when the temperature is raised to \(30^{\circ} \mathrm{C}\). Find \(\alpha\). (b) If an aluminum pole is \(180 \mathrm{~cm}\) long at \(15^{\circ} \mathrm{C}\), how long is the pole if the temperature is raised to \(40^{\circ} \mathrm{C}\) ? [Take \(\left.\alpha=2.3 \times 10^{-5} /{ }^{\circ} \mathrm{C} .\right]\)

Find the limits. $$ \lim _{x \rightarrow \pi}(x-\pi) \cot x $$

(a) Let \(y=\sqrt{x}\). Find \(d y\) and \(\Delta y\) at \(x=9\) with \(d x=\Delta x=-1\) (b) Sketch the graph of \(y=\sqrt{x}\), showing \(d y\) and \(\Delta y\) in the picture.
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