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Chapter 6: Problem 40

Find all functions \(f(t)\) with the following property: $$f^{\prime}(t)=t^{2}-5 t-7$$

Short Answer

Expert verified
The functions have the form \(f(t) = \frac{t^3}{3} - \frac{5t^2}{2} - 7t + C\).

Step by step solution

01

Understand the Problem

We need to find a function, referred to as \(f(t)\), where its derivative \(f^{\r_prime}(t)\) satisfies the equation \(f^{\r_prime}(t) = t^2 - 5t - 7\). This requires us to integrate the given derivative.
02

Integrate the Given Derivative

Integrate both sides of the equation \(f^{\r_prime}(t) = t^2 - 5t - 7\). Start by setting up the integral: \[f(t) = \r\ral\ra (t^2 - 5t - 7) \r\rt\ra + C\] where \(C\) is the constant of integration.
03

Perform the Integration

Calculate the integral by integrating each term separately: \[ \r\ral t^2 \rt\ra dt = \frac{t^3}{3} \] \[ \r\ral -5t \rt\ra dt = -\frac{5t^2}{2} \] \[ \r\ral -7 \rt\ra dt = -7t \]
04

Combine the Results

Combine the results of the individual integrals to form the final expression: \[ f(t) = \frac{t^3}{3} - \frac{5t^2}{2} - 7t + C \]

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Key Concepts

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

Finding Antiderivatives
To understand how to find antiderivatives, remember that they are the reverse process of differentiation. The goal is to determine a function whose derivative matches a given function. In our exercise, we want to find a function, specifically an antiderivative, that satisfies the equation:
\(f'(t) = t^2 - 5t - 7\).
We achieve this by integrating each term of the derivative. Here are the steps to follow:
  • Set up the integral: \int (t^2 - 5t - 7) dt\.
  • Integrate term-by-term:
    • The integral of \(t^2\) is \frac{t^3}{3}\.
    • The integral of \(-5t\) is \frac{-5t^2}{2}\.
    • The integral of \(-7\) is \-7t\.
After integrating, combine the results:
\ f(t) = \frac{t^3}{3} - \frac{5t^2}{2} - 7t + C \ where \(C\) is the constant of integration.
Constant of Integration
When finding antiderivatives, a crucial part is the constant of integration, represented as \(C\). It arises because differentiation of a constant is zero, meaning we lose this constant information during differentiation. Thus, any antiderivative must include an unspecified constant.
The general solution to an indefinite integral includes this constant \(C\), ensuring we account for all possible original functions. For our example, the solution is:
\ f(t) = \frac{t^3}{3} - \frac{5t^2}{2} - 7t + C \ This \(C\) means there are infinitely many functions satisfying the equation. No additional information can help determine its value precisely.
Polynomial Functions
A polynomial function consists of terms of the form \(a_n t^n\) where \(a_n\) are coefficients and \(n\) are non-negative integers. Each term is a product of a coefficient and a power of the variable. In our example, the polynomial is \(t^2 - 5t - 7\). Integrating a polynomial means finding an antiderivative for each term:
  • The integral of \(t^2\) is \frac{t^3}{3}\, increasing the power by one and dividing by the new power.
  • The integral of \(-5t\) is \-5 \times \frac{t^2}{2}\.
  • The integral of the constant \(-7\) is \-7t\.
The resulting polynomial from the integration process is:
\ f(t) = \frac{t^3}{3} - \frac{5t^2}{2} - 7t + C \ Understanding how polynomials integrate helps with handling more complex functions and extending knowledge to higher-level calculus topics.

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Most popular questions from this chapter

Suppose that the interval \(0 \leq x \leq 1\) is divided into 100 subintervals of width \(\Delta x=.01\). Show that the following sum is close to \(5 / 4\). $$ \begin{aligned} \left[2(.01)+(.01)^{3}\right] \Delta x &+\left[2(.02)+(.02)^{3}\right] \Delta x \\ &+\cdots+\left[2(1.0)+(1.0)^{3}\right] \Delta x \end{aligned} $$ The following exercises ask for an unknown quantity \(x\). After setting up the appropriate formula involving a definite integral, use the fundamental theorem to evaluate the definite integral as an expression in \(x\). Because the resulting equation will be too complicated to solve algebraically, you must use a graphing utility to obtain the solution. (Note: If the quantity \(x\) is an interest rate paid by a savings account, it will most likely be between 0 and . 10.)

Find the area of the region between the curve and the \(x\) -axis. \(f(x)=1-x^{2}\), from \(-2\) to 2.

Find the area of the region bounded by the curves. \(y=x^{2}-1\) and \(y=3\)

Find the area of the region between the curves. \(y=e^{x}\) and \(y=\frac{1}{x^{2}}\) from \(x=1\) to \(x=2\)

Evaluate a Riemann sum to approximate the area under the graph of \(f(x)\) on the given interval, with points selected as specified. \(f(x)=\sqrt{1-x^{2}} ;-1 \leq x \leq 1, n=20\), left endpoints of subintervals
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