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Chapter 2: Problem 127

Consider a function \(f(x)\) and its derivative \(d f l d x\). Does this derivative have to be a function of \(x\) ?

Short Answer

Expert verified
Answer: The derivative of a function is usually related to the independent variable x, and it will be a function of x if it is defined for all values of x in the domain of the original function. However, there could be cases where the derivative might not exist for certain values of x, making it a partial function of x instead.

Step by step solution

01

Recall the definition of a derivative

A derivative of a function, \(f(x)\), is a measure of the rate of change of the function with respect to the independent variable \(x\). The derivative is denoted by \(\frac{d f}{d x}\) or \(f'(x)\). It is defined as the limit: $$ \frac{d f}{d x} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$ where \(h\) is a very small change in the independent variable \(x\).
02

Analyze the relationship between the function and its derivative

The derivative of a function is itself a function if it is defined for all values of \(x\) in the domain of \(f(x)\). However, there could be cases where the derivative might exist for some values of \(x\) but not the others. In those cases, the derivative will not be a function of \(x\) in the traditional sense but still can be considered a partial function.
03

Provide examples

(a) Consider the function \(f(x) = x^2\). Its derivative is given by: $$ \frac{d f}{d x} = 2x $$ This derivative is a function of \(x\), as it is defined for all values of \(x\). (b) Consider the function \(g(x) = |x|\) where \(|x|\) denotes the absolute value of \(x\). This function looks like a "V" shape with a sharp point at \(x=0\). At the point \(x=0\), the function does not have a well-defined tangent, and therefore, the derivative does not exist at \(x=0\). In this case, the derivative of \(g(x)\) exists for all \(x\) except \(x=0\), so it can be considered a partial function of \(x\).
04

Conclusion

The derivative of a function, \(f(x)\), will be a function of \(x\) if it is defined for all values of \(x\) in the domain of \(f(x)\). In some cases, the derivative might not exist for certain values of \(x\), making it a partial function of \(x\). However, in general, the derivative can be considered related to the variable \(x\).

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Most popular questions from this chapter

Is the thermal conductivity of a medium, in general, constant or does it vary with temperature?

Consider a 20-cm-thick large concrete plane wall \((k=0.77 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) subjected to convection on both sides with \(T_{\infty 1}=27^{\circ} \mathrm{C}\) and \(h_{1}=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the inside, and \(T_{\infty 2}=8^{\circ} \mathrm{C}\) and \(h_{2}=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the outside. Assuming constant thermal conductivity with no heat generation and negligible radiation, (a) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the wall, (b) obtain a relation for the variation of temperature in the wall by solving the differential equation, and \((c)\) evaluate the temperatures at the inner and outer surfaces of the wall.

A flat-plate solar collector is used to heat water by having water flow through tubes attached at the back of the thin solar absorber plate. The absorber plate has an emissivity and an absorptivity of \(0.9\). The top surface \((x=0)\) temperature of the absorber is \(T_{0}=35^{\circ} \mathrm{C}\), and solar radiation is incident on the absorber at \(500 \mathrm{~W} / \mathrm{m}^{2}\) with a surrounding temperature of \(0^{\circ} \mathrm{C}\). Convection heat transfer coefficient at the absorber surface is \(5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while the ambient temperature is \(25^{\circ} \mathrm{C}\). Show that the variation of temperature in the absorber plate can be expressed as \(T(x)=-\left(\dot{q}_{0} / k\right) x+T_{0}\), and determine net heat flux \(\dot{q}_{0}\) absorbed by the solar collector.

Consider steady one-dimensional heat conduction in a plane wall in which the thermal conductivity varies linearly. The error involved in heat transfer calculations by assuming constant thermal conductivity at the average temperature is \((a)\) none, \((b)\) small, or \((c)\) significant.

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) \(T(x)=40 x-80\)
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