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Chapter 3: Problem 63

Determining the unknown constant Let $$ f(x)=\left\\{\begin{array}{ll} 2 x^{2} & \text { if } x \leq 1 \\ a x-2 & \text { if } x > 1 \end{array}\right. $$ Determine a value of \(a\) (if possible) for which \(f^{\prime}\) is continuous at \(x=1\)

Short Answer

Expert verified
Answer: The value of \(a\) that makes the function continuous is \(a=4\).

Step by step solution

01

Finding the derivative of the first part of the function

To find the derivative of the part \(2x^2\), we'll apply the power rule: $$ f'(x)= \frac{d}{dx}(2x^2) = 2(2x) = 4x $$
02

Finding the derivative of the second part of the function

To find the derivative of the part \(ax-2\), we'll do it term by term: $$ f'(x)= \frac{d}{dx}(ax-2) = a $$ The first derivative is a constant function.
03

Checking continuity condition

To make sure the derivative is continuous, we'll make sure both parts have the same value at \(x=1\). That is: $$ f'(1)=4(1)=4 $$ Now we need to find the value of \(a\) that makes its part of the function equal to 4 at \(x=1\). Since \(f'(x)=a\): $$ a=4 $$
04

Conclusion

The value of \(a\) that makes the derivative continuous at \(x=1\) is \(a=4\).

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Key Concepts

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

Power Rule Differentiation
Understanding the power rule for differentiation is fundamental for students tackling calculus problems. The rule is simply stated: for any real number power, say n, the derivative of x^n with respect to x is nx^(n-1).

This rule is a cornerstone in finding derivatives as it applies to monomials where n can be any real number, positive or negative (except where x = 0 and n is negative, which is not defined). It is also utilized as part of the derivative process for polynomials, which are sums of monomials.

For example, in our exercise, to find the derivative of 2x^2, the rule tells us to bring down the exponent (which is 2) to the front, multiply by the coefficient (which is 2), and then decrease the exponent by one. Mathematically, this is shown as follows:
\[ f'(x) = \frac{d}{dx}(2x^2) = 2 \cdot 2x^{(2-1)} = 4x \]
In a more general application, this rule saves time and reduces complexity when dealing with higher powers, making the process of differentiation straightforward and hence easier for students to grasp.
Piecewise Function
A piecewise function is a mathematical function which is defined by multiple sub-functions, each applying to a certain interval of the main function's domain. These sub-functions may be different expressions, and they often correspond to physical or conceptual separate 'pieces' of the domain.

Our given exercise features a piecewise function which is a classic example:
\[ f(x)=\left\{\begin{array}{ll} 2 x^{2} & \text { if } x \leq 1 \ a x-2 & \text { if } x > 1 \end{array}\right. \]
The function has two expressions: 2x^2 applies when x is less than or equal to 1, and ax-2 when x is greater than 1. A piecewise function is continuous at a point if the limit from the left and the limit from the right at that point are equal. The behavior at the 'join' points where the formulas change is particularly important when looking at the continuity and differentiability of the function.
Determining Constants in Functions
In some mathematical problems involving functions, we encounter unknown constants like a in the function expression. Determining the value of such constants often involves understanding the behavior of the function, specifically in terms of continuity or differentiability.

In our current exercise, we were tasked to find a value for the constant a that ensures the function's derivative is continuous at a particular point, x=1. This involves setting up an equation where the derivatives from both 'pieces' of the function are equal at that point:
\[ f'(1)=4(1)=4 \]
On solving the equation for the unknown constant, we find a such that:
\[ a = f'(1) = 4 \]
This step is crucial because it ensures the smoothness of the graph of f(x) at x=1 and guarantees that the function behaves nicely without any jumps or breaks. Exercises like these are important for developing understanding in areas ranging from physics to economics, where ensuring continuity of functions is pivotal.

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Most popular questions from this chapter

The volume of a torus (doughnut or bagel) with an inner radius of \(a\) and an outer radius of \(b\) is \(V=\pi^{2}(b+a)(b-a)^{2} / 4\) a. Find \(d b / d a\) for a torus with a volume of \(64 \pi^{2}\). b. Evaluate this derivative when \(a=6\) and \(b=10\)

Suppose a large company makes 25,000 gadgets per year in batches of \(x\) items at a time. After analyzing setup costs to produce each batch and taking into account storage costs, it has been determined that the total cost \(C(x)\) of producing 25,000 gadgets in batches of \(x\) items at a time is given by $$C(x)=1,250,000+\frac{125,000,000}{x}+1.5 x.$$ a. Determine the marginal cost and average cost functions. Graph and interpret these functions. b. Determine the average cost and marginal cost when \(x=5000\). c. The meaning of average cost and marginal cost here is different from earlier examples and exercises. Interpret the meaning of your answer in part (b).

The flow of a small stream is monitored for 90 days between May 1 and August 1. The total water that flows past a gauging station is given by $$V(t)=\left\\{\begin{array}{ll}\frac{4}{5} t^{2} & \text { if } 0 \leq t<45 \\\\-\frac{4}{5}\left(t^{2}-180 t+4050\right) & \text { if } 45 \leq t<90, \end{array}\right.$$ where \(V\) is measured in cubic feet and \(t\) is measured in days, with \(t=0\) corresponding to May 1. a. Graph the volume function. b. Find the flow rate function \(V^{\prime}(t)\) and graph it. What are the units of the flow rate? c. Describe the flow of the stream over the 3 -month period. Specifically, when is the flow rate a maximum?

Identifying functions from an equation. The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. $$x^{4}=2\left(x^{2}-y^{2}\right) \text { (eight curve) }$$

Tangency question It is easily verified that the graphs of \(y=x^{2}\) and \(y=e^{x}\) have no points of intersection (for \(x>0\) ), and the graphs of \(y=x^{3}\) and \(y=e^{x}\) have two points of intersection. It follows that for some real number \(2 0 \) ). Using analytical and/or graphical methods, determine \(p\) and the coordinates of the single point of intersection.
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