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Chapter 6: Problem 9

(Maple) (a) Plot a graph of \(y=\frac{1}{\sqrt{x}}\) on \(0.01 \leq x \leq 1\). Evaluate \(\int_{N}^{1} \frac{1}{\sqrt{x}} \mathrm{~d} x\) in the cases when \(N\) is \(0.1,0.01\) and \(0.001\) What do these results suggest about the value of \(\int_{0}^{1} \frac{1}{\sqrt{x}} \mathrm{~d} x ?\) (b) Repeat part (a) for the function \(y=\frac{1}{x^{2}}\).

Short Answer

Expert verified
For Part (a), the integral approaches 2 as N approaches 0. For Part (b), the integral diverges to infinity.

Step by step solution

01

Plot the graph for Part (a)

Plot the graph of the function \[y = \frac{1}{\sqrt{x}}\] on the interval \[0.01 \leq x \leq 1\]. You can use graphing software such as Maple, Desmos, or an online graphing tool.
02

Compute the integral for Part (a) at different N values

Evaluate the integral \[\int_{N}^{1} \frac{1}{\sqrt{x}} \mathrm{~d} x\] for \[N = 0.1, 0.01, \text{and} 0.001\]. The integral can be evaluated as: \[\int \frac{1}{\sqrt{x}} \mathrm{~d} x = 2 \sqrt{x} + C\]. Substitute the limits to find: \[\int_{N}^{1} \frac{1}{\sqrt{x}} \mathrm{~d} x = \left[ 2 \sqrt{x} \right]_{N}^{1} = 2 - 2 \sqrt{N}\]. For each given \(N\), calculate the value.
03

Interpret the results for Part (a)

Calculate the results: \(N = 0.1\): \[2 - 2 \sqrt{0.1} \approx 1.367\], \(N = 0.01\): \[2 - 2 \sqrt{0.01} = 1.8\], and \(N = 0.001\): \[2 - 2 \sqrt{0.001} \approx 1.936\]. As \(N\) approaches 0, the values of the integral approach 2. This suggests that \[\int_{0}^{1} \frac{1}{\sqrt{x}} \mathrm{~d} x\] is likely 2.
04

Plot the graph for Part (b)

Plot the graph of the function \[y = \frac{1}{x^{2}}\] on the interval \[0.01 \leq x \leq 1\]. Use graphing software to visualize the relationship.
05

Compute the integral for Part (b) at different N values

Evaluate the integral \[\int_{N}^{1} \frac{1}{x^{2}} \mathrm{~d} x\] for \[N = 0.1, 0.01, \text{and} 0.001\]. The integral is computed as: \[\int \frac{1}{x^{2}} \mathrm{~d} x = - \frac{1}{x} + C\]. Substitute the limits to find: \[\int_{N}^{1} \frac{1}{x^{2}} \mathrm{~d} x = \left[ - \frac{1}{x} \right]_{N}^{1} = -1 + \frac{1}{N}\]. Calculate the value for each given \(N\).
06

Interpret the results for Part (b)

Calculate the results: \(N = 0.1\): \[-1 + \frac{1}{0.1} = 9\], \(N = 0.01\): \[-1 + \frac{1}{0.01} = 99\], and \(N = 0.001\): \[-1 + \frac{1}{0.001} = 999\]. As \(N\) approaches 0, the values of the integral increase without bound. This suggests that \[\int_{0}^{1} \frac{1}{x^{2}} \mathrm{~d} x\] diverges to infinity.

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Key Concepts

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

Improper Integrals
Improper integrals deal with integrals that have infinite limits or integrands with infinite discontinuities. Often, these integrals appear in scenarios where the function behaves badly near the boundaries. For instance, when evaluating \(\text{int_{0}^{1} \frac{1}{\sqrt{x}} \text{~d} x}\).
Here are the steps to handle such integrals:
  • Identify points where the function is undefined or where the limits are infinite.
  • Split the integral at the point of discontinuity or change limits to a finite number.
  • Evaluate the limit of the integral as the problematic point is approached.
It's crucial to determine if the integral converges to a finite value or diverges to infinity by examining the behavior as it approaches the boundaries.
Definite Integrals
Definite integrals are used to find the area under a curve between two specific points. They are evaluated using the Fundamental Theorem of Calculus. For instance, in our example
\(\text{int_{N}^{1} \frac{1}{x^2} \text{~d} x = -\frac{1}{x} + C}\)
you would follow these steps:
  • Find the antiderivative of the function.
  • Evaluate the antiderivative at the upper and lower limits.
  • Subtract the value at the lower limit from the value at the upper limit.
One key distinction is that, unlike improper integrals, definite integrals have specific boundaries and usually result in a finite area. However, sometimes when the boundaries extend to infinity, we move into the realm of improper integrals.
Graphing Functions
Graphing functions visually represents mathematical equations, providing insights into their behavior. To graph a function like \(\text{y = \frac{1}{\sqrt{x}}}\) on an interval, take these steps:

Use graphing software or tools like Maple or Desmos.
  • Input the equation and specify the interval, e.g., \(\text{0.01 \leq x \leq 1}\).
  • Observe how the curve behaves within this range.
Simplifying the process helps to understand the nature of the function. Additionally, a visual graph easily highlights areas where the function might become undefined or approach infinity, crucial for solving integral calculus questions. This visualization plays a vital role in interpreting and solving the integral problems at hand.
Asymptotic Behavior
Asymptotic behavior refers to how a function behaves as it approaches a specific value or infinity. For example, consider the function \(\text{y = \frac{1}{x^2}}\):

As \(\text{x approaches 0, y }\) increases without bound.
  • For \(\text{x > 1}\), the graph gradually approaches the x-axis, suggesting that \(\text{y }\) approaches 0 as \(\text{x increases}\).
  • Understanding these behaviors can predict the convergence or divergence of integrals.
Analyzing this behavior is crucial when dealing with improper integrals. By evaluating limits and approaching critical points, we can determine the nature of the function and thus the area under the curve.

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Most popular questions from this chapter

The present value of a continuous revenue stream of \(\$ 5000\) per year with a discount rate of \(10 \%\) over \(n\) years is \(\$ 25000\). Find the value of \(n\) correct to 1 decimal place.

Calculate the present value of a continuous revenue stream of \(\$ 1000\) per year if the discount rate is \(5 \%\) and the money is paid (a) for 3 years (b) for 10 years (c) for 100 years (d) in perpetuity

Evaluate the following definite integrals: (a) \(\int_{0}^{1} x^{3} \mathrm{~d} x\) (b) \(\int_{2}^{5}(2 x-1) \mathrm{d} x\) (c) \(\int_{1}^{4}\left(x^{2}-x+1\right) \mathrm{d} x\) (d) \(\int_{0}^{1} e^{x} \mathrm{~d} x\)

(a) A firm's marginal cost function is $$ \mathrm{MC}=2 $$ Find an expression for the total cost function if the fixed costs are 500 . Hence find the total cost of producing 40 goods. (b) The marginal revenue function of a monopolistic producer is $$ \mathrm{MR}=100-6 Q $$ Find the total revenue function and deduce the corresponding demand equation. (c) Find an expression for the savings function if the marginal propensity to save is given by $$ \mathrm{MPS}=0.4-0.1 Y^{-1 / 2} $$ and savings are zero when income is 100 .

Find the area under the graph of $$ f(x)=4 x^{3}-3 x^{2}+4 x+2 $$ between \(x=1\) and \(x=2\).
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