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Chapter 4: Problem 19

Find the most general antiderivative of the function.(Check your answer by differentiation.) \(f(t)=\frac{t^{4}-t^{2}+1}{t^{2}}\)

Short Answer

Expert verified
The antiderivative is \(F(t) = \frac{t^3}{3} - t - \frac{1}{t} + C\).

Step by step solution

01

Simplify the Function

Given the function is \(f(t) = \frac{t^4 - t^2 + 1}{t^2}\). Start by dividing each term in the numerator by \(t^2\). This will give:\[f(t) = t^2 - 1 + \frac{1}{t^2}\]
02

Integrate Term-by-Term

Now, find the antiderivative by integrating each term separately:1. \( \int t^2 \, dt = \frac{t^3}{3} + C_1 \)2. \( \int (-1) \, dt = -t + C_2 \)3. \( \int t^{-2} \, dt = -t^{-1} + C_3 \)Combine the constants into a single constant \(C\):\[\int f(t) \, dt = \frac{t^3}{3} - t - \frac{1}{t} + C\]
03

Differentiate to Check

To verify, differentiate \(F(t) = \frac{t^3}{3} - t - \frac{1}{t} + C\). Use the power rule for differentiation:1. \( \frac{d}{dt} \left( \frac{t^3}{3} \right) = t^2 \)2. \( \frac{d}{dt} (-t) = -1 \)3. \( \frac{d}{dt} (-\frac{1}{t}) = \frac{1}{t^2} \)Combine these results:\[ \frac{d}{dt} \left( \frac{t^3}{3} - t - \frac{1}{t} + C \right) = t^2 - 1 + \frac{1}{t^2} = f(t) \]Thus, the differentiated function equals the original function.

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Key Concepts

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

Integration Techniques
Integration is the process of finding the antiderivative of a function, which is essentially the reverse of differentiation. In this exercise, we aim to find the most general antiderivative of the given function. A key integration technique utilized here is integrating a function term-by-term. Breaking down the integral into simpler components lets us apply basic integration rules to each term individually.
  • First, simplify the function if necessary, allowing for easier integration.
  • Separate the function into individual parts that can be integrated independently.
  • Combine the results at the end, including a constant of integration, since the antiderivative is any function whose derivative is the given function.

Each part of the exercise follows these principles. Simplifying the expression \( \frac{t^{4}-t^{2}+1}{t^{2}} \) allows for easier term-by-term integration and illustrates the beauty of integration techniques.
Power Rule
The power rule is a fundamental technique in both differentiation and integration. It allows us to handle functions that are in the form of power expressions. When integrating, the power rule states that the antiderivative of a function \( t^n \) is given by \( \frac{t^{n+1}}{n+1} \), provided \( n eq -1 \).

Here鈥檚 how it applies to our exercise:
  • For \( \int t^2 \, dt \), we increase the power by 1, resulting in \( \frac{t^3}{3} \).
  • When dealing with \( t^{-2} \), the antiderivative is \( \int t^{-2} \, dt = -t^{-1} \), which also uses the power rule.
  • Notice how integrating each term applies the power rule individually, making the process systematic and predictable.

Understanding and applying the power rule is crucial, as it's both a powerful tool and a stepping stone to more complex integration techniques.
Simplifying Rational Expressions
Rational expressions involve polynomials in the numerator and the denominator. Simplifying these expressions is often necessary before integration or differentiation. In our exercise, we started with \( \frac{t^{4}-t^{2}+1}{t^{2}} \). Simplification is key here:
  • Divide each term of the polynomial in the numerator by the polynomial in the denominator.
  • This changes the function into a form that is easier to integrate, specifically splitting into three separate terms: \( t^2 - 1 + \frac{1}{t^2} \).
  • Each of these terms can now be tackled independently, simplifying the integration process.

Understanding the process of simplifying rational expressions allows us to transform complex expressions into simple, manageable forms. This step is often the gateway to applying straightforward integration techniques and obtaining the solution.

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Most popular questions from this chapter

Crows and whelks Crows on the west coast of Canada feed on whelks by carrying them to heights of about 5 m and dropping them onto rocks (several times if necessary) to break open their shells. Two of the questions raised by the author of a study of this phenomenon were "Do crows drop whelks from the best height for breaking?" and "How energetically profitable is dropping of whelks?" The author constructed poles and dropped whelks from various heights. A model based on the study's data for the number of times a whelk needs to be dropped from a height h to be broken is \(n(h)=\frac{h+14.8}{h-1.2}\) where \(h\) is measured in meters. The energy expended by a crow in this activity is proportional to the height \(h\) and to the number \(n(h) :\) \(E=k h n(h)=\frac{k h(h+14.8)}{h-1.2}\) (a) What value of \(h\) minimizes the energy expended by the crows? (b) How does your answer to part (a) compare with the observed average dropping height of 5.3 m that is actually used by crows? Does the model support the existence of an optimal foraging strategy?

Find the equilibria of the difference equation and classify them as stable or unstable. Use cobwebbing to find \(\lim _{t \rightarrow \infty} x_{t}\) for the given initial values. \(x_{t+1}=\frac{7 x_{t}^{2}}{x_{t}^{2}+10}, \quad x_{0}=1, \quad x_{0}=3\)

Find \(f\) . \(f^{\prime \prime}(x)=8 x^{3}+5, \quad f(1)=0, \quad f^{\prime}(1)=8\)

\(37-44\) (a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts (a)-(d) to sketch the graph of \(f .\) $$f(x)=\sqrt{x^{2}+1}-x$$

Find \(f\) . \(f^{\prime}(x)=\sqrt{x}(6+5 x), \quad f(1)=10\)
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