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Chapter 2: Problem 130

What is the difference between the degree and the order of a derivative?

Short Answer

Expert verified
Answer: The degree of a derivative refers to the highest power in the polynomial equation, which determines the overall behavior of the function. The order of a derivative refers to the number of times a function has been differentiated, revealing information about the function's rate of change. Degree and order are essential concepts for understanding and analyzing functions in calculus.

Step by step solution

01

Understanding Degree #

The degree of a derivative refers to the highest power in the polynomial equation. In simple words, it's the exponent of the highest-degree term. A derivative's degree gives information about the behavior of the function, like the number of potential maximums, minimums, and inflection points. A polynomial of degree "n" will have "n" zeroes. For example, consider the function f(x) = 3x^5 - 2x^3 + x^2 - 1. The highest power in this function is 5, so the degree of f(x) is 5.
02

Understanding Order #

The order of a derivative refers to the number of times a function has been differentiated. So, the first derivative is the first-order derivative, the second derivative is the second-order derivative, and so on. The order of a derivative tells us the rate at which the function is changing. A higher-order derivative tells us how the function's rate of change changes itself. For the example function f(x) = 3x^5 - 2x^3 + x^2 -1, - First derivative, f'(x) = 15x^4 - 6x^2 + 2x, (order = 1) - Second derivative, f''(x) = 60x^3 - 12x + 2, (order = 2)
03

Difference between Degree and Order #

Now that we're familiar with the terms "degree" and "order," let's discuss the key differences between them: 1. Degree refers to the highest power in the polynomial, whereas order refers to the number of times a function has been differentiated. 2. The degree determines the overall behavior of the function, while the order reveals information about the function's rate of change. 3. A higher degree does not necessarily imply a higher order, and vice versa. In summary, the degree of a derivative specifies the highest power of the function, while the order of a derivative refers to the number of times that the function has been differentiated. Both concepts are essential for understanding and analyzing functions in calculus.

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Most popular questions from this chapter

Consider a homogeneous spherical piece of radioactive material of radius \(r_{o}=0.04 \mathrm{~m}\) that is generating heat at a constant rate of \(\dot{e}_{\mathrm{gen}}=4 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\). The heat generated is dissipated to the environment steadily. The outer surface of the sphere is maintained at a uniform temperature of \(80^{\circ} \mathrm{C}\), and the thermal conductivity of the sphere is $k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(. Assuming steady one-dimensional heat transfer, \)(a)$ express the differential equation and the boundary conditions for heat conduction through the sphere, \((b)\) obtain a relation for the variation of temperature in the sphere by solving the differential equation, and \((c)\) determine the temperature at the center of the sphere.

Consider a solid cylindrical rod whose ends are maintained at constant but different temperatures while the side surface is perfectly insulated. There is no heat generation. It is claimed that the temperature along the axis of the rod varies linearly during steady heat conduction. Do you agree with this claim? Why?

Consider a third-order linear and homogeneous differential equation. How many arbitrary constants will its general solution involve?

A spherical container with an inner radius \(r_{1}=1 \mathrm{~m}\) and an outer radius \(r_{2}=1.05 \mathrm{~m}\) has its inner surface subjected to a uniform heat flux of \(\dot{q}_{1}=7 \mathrm{~kW} / \mathrm{m}^{2}\). The outer surface of the container has a temperature \(T_{2}=25^{\circ} \mathrm{C}\), and the container wall thermal conductivity is $k=1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Show that the variation of temperature in the container wall can be expressed as $T(r)=\left(\dot{q}_{1} r_{1}^{2} / k\right)\left(1 / r-1 / r_{2}\right)+T_{2}$ and determine the temperature of the inner surface of the container at \(r=r_{1}\).

A plane wall of thickness \(L\) is subjected to convection at both surfaces with ambient temperature \(T_{\infty \text { el }}\) and heat transfer coefficient \(h_{1}\) at the inner surface and corresponding \(T_{\infty 2}\) and \(h_{2}\) values at the outer surface. Taking the positive direction of \(x\) to be from the inner surface to the outer surface, the correct expression for the convection boundary condition is (a) \(k \frac{d T(0)}{d x}=h_{1}\left[T(0)-T_{\infty 1}\right]\) (b) $k \frac{d T(L)}{d x}=h_{2}\left[T(L)-T_{\infty 22}\right]$ (c) $-k \frac{d T(0)}{d x}=h_{1}\left(T_{\infty 1}-T_{\infty 22}\right)(d)-k \frac{d T(L)}{d x}=h_{2}\left(T_{\infty \infty 1}-T_{\infty 22}\right)$ (e) None of them
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