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Chapter 2: Problem 124

What is the geometrical interpretation of a derivative? What is the difference between partial derivatives and ordinary derivatives?

Short Answer

Expert verified
Short Answer: The geometrical interpretation of a derivative is the slope of the tangent line to a function's graph at a given point, representing the rate of change of the function at that point. Ordinary derivatives are used for functions with one variable and are denoted by \(f'(x)\) or \(\frac{dy}{dx}\). Partial derivatives are used for multivariable functions and are denoted by \(\frac{\partial f}{\partial x}\). The main difference between them is that ordinary derivatives represent the rate of change concerning the single input variable, while partial derivatives represent the rate of change of a multivariable function with respect to one variable, keeping the other variables constant.

Step by step solution

01

Geometrical Interpretation of a Derivative

The derivative of a function has a geometrical interpretation: it represents the slope of the tangent line to the function's graph at a given point. In other words, the derivative tells us how steep the function is at a particular point, by providing the ratio of the change in the y-axis (vertical) to the change in the x-axis (horizontal). This can help us understand the rate of change of a function at a specific point.
02

Ordinary Derivatives

An ordinary derivative, often written as \(f'(x)\) or \(\frac{dy}{dx}\), is used when we have a function of one variable, i.e., a function that has only one input. Areas where we use ordinary derivatives include measuring the velocity of a moving object, finding the maximum or minimum values of a function, and analyzing rates of change in various situations.
03

Partial Derivatives

Partial derivatives come into play when we have multivariable functions, i.e., functions that depend on more than one input variable. The partial derivative of a function focuses on how the function changes concerning one variable while keeping the other variables constant. We denote a partial derivative with respect to a variable by ∂ instead of d in the derivative notation (e.g., \(\frac{\partial f}{\partial x}\) represents the partial derivative of function f with respect to x).
04

Difference between Partial and Ordinary Derivatives

The main differences between partial derivatives and ordinary derivatives are: 1. Use: Ordinary derivatives are used for functions with one variable, while partial derivatives are used for multivariable functions. 2. Notation: Ordinary derivatives are denoted by \(f'(x)\) or \(\frac{dy}{dx}\), while partial derivatives are denoted by \(\frac{\partial f}{\partial x}\), where x is the variable we differentiate with respect to. 3. Interpretation: Ordinary derivatives represent the rate of change of a function concerning the single input variable, while partial derivatives represent the rate of change of a multivariable function with respect to one variable, keeping the other variables constant.

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Most popular questions from this chapter

Consider a steam pipe of length \(L=35 \mathrm{ft}\), inner radius \(r_{1}=2\) in, outer radius \(r_{2}=2.4\) in, and thermal conductivity \(k=8 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\). Steam is flowing through the pipe at an average temperature of \(250^{\circ} \mathrm{F}\), and the average convection heat transfer coefficient on the inner surface is given to be \(h=\) \(15 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\). If the average temperature on the outer surfaces of the pipe is \(T_{2}=160^{\circ} \mathrm{F},(a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the pipe, \((b)\) obtain a relation for the variation of temperature in the pipe by solving the differential equation, and \((c)\) evaluate the rate of heat loss from the steam through the pipe.

The temperatures at the inner and outer surfaces of a 15 -cm-thick plane wall are measured to be \(40^{\circ} \mathrm{C}\) and \(28^{\circ} \mathrm{C}\), respectively. The expression for steady, one-dimensional variation of temperature in the wall is (a) \(T(x)=28 x+40\) (b) \(T(x)=-40 x+28\) (c) \(T(x)=40 x+28\) (d) \(T(x)=-80 x+40\) (e) \(T(x)=40 x-80\)

Consider a \(1.5\)-m-high and \(0.6-\mathrm{m}\)-wide plate whose thickness is \(0.15 \mathrm{~m}\). One side of the plate is maintained at a constant temperature of \(500 \mathrm{~K}\) while the other side is maintained at \(350 \mathrm{~K}\). The thermal conductivity of the plate can be assumed to vary linearly in that temperature range as \(k(T)=\) \(k_{0}(1+\beta T)\) where \(k_{0}=18 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\beta=8.7 \times 10^{-4} \mathrm{~K}^{-1}\). Disregarding the edge effects and assuming steady onedimensional heat transfer, determine the rate of heat conduction through the plate. Answer: \(22.2 \mathrm{~kW}\)

Hot water flows through a PVC \((k=0.092 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) pipe whose inner diameter is \(2 \mathrm{~cm}\) and outer diameter is \(2.5 \mathrm{~cm}\). The temperature of the interior surface of this pipe is \(50^{\circ} \mathrm{C}\) and the temperature of the exterior surface is \(20^{\circ} \mathrm{C}\). The rate of heat transfer per unit of pipe length is (a) \(77.7 \mathrm{~W} / \mathrm{m}\) (b) \(89.5 \mathrm{~W} / \mathrm{m}\) (c) \(98.0 \mathrm{~W} / \mathrm{m}\) (d) \(112 \mathrm{~W} / \mathrm{m}\) (e) \(168 \mathrm{~W} / \mathrm{m}\)

Exhaust gases from a manufacturing plant are being discharged through a 10 - \(\mathrm{m}\) tall exhaust stack with outer diameter of \(1 \mathrm{~m}\), wall thickness of \(10 \mathrm{~cm}\), and thermal conductivity of \(40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The exhaust gases are discharged at a rate of \(1.2 \mathrm{~kg} / \mathrm{s}\), while temperature drop between inlet and exit of the exhaust stack is \(30^{\circ} \mathrm{C}\), and the constant pressure specific heat of the exhaust gasses is \(1600 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). On a particular day, the outer surface of the exhaust stack experiences radiation with the surrounding at \(27^{\circ} \mathrm{C}\), and convection with the ambient air at \(27^{\circ} \mathrm{C}\) also, with an average convection heat transfer coefficient of \(8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Solar radiation is incident on the exhaust stack outer surface at a rate of \(150 \mathrm{~W} / \mathrm{m}^{2}\), and both the emissivity and solar absorptivity of the outer surface are 0.9. Assuming steady one-dimensional heat transfer, (a) obtain the variation of temperature in the exhaust stack wall and (b) determine the inner surface temperature of the exhaust stack.
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