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Chapter 5: Problem 77

Suppose that \(\int_{1}^{x} f(t) d t=x^{2}-2 x+1 .\) Find \(f(x)\).

Short Answer

Expert verified
The function \(f(x) = 2x - 2\).

Step by step solution

01

Understand the Problem Statement

We are given a definite integral \(\int_{1}^{x} f(t) \, dt = x^2 - 2x + 1\). This integral states that when you integrate the function \(f(t)\) from 1 to \(x\), you obtain the expression \(x^2 - 2x + 1\). Our task is to determine the function \(f(x)\).
02

Recall the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus tells us that if \(F(x) = \int_{a}^{x} f(t) \, dt\), then the derivative of \(F(x)\) with respect to \(x\) gives \(f(x)\). Therefore, we can write \(f(x) = \frac{d}{dx}\left(\int_{1}^{x} f(t) \, dt\right)\).
03

Differentiate the Right Side

According to the problem, \(\int_{1}^{x} f(t) \, dt = x^2 - 2x + 1\). We need to differentiate the right-hand side with respect to \(x\) to find \(f(x)\). Differentiating each term, we get: \(\frac{d}{dx}(x^2 - 2x + 1) = 2x - 2\).
04

Conclusion

From differentiating \(x^2 - 2x + 1\), we found that \(f(x) = 2x - 2\). Thus, the function \(f(x)\) is \(2x - 2\).

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Key Concepts

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

Definite Integral
A definite integral is a fundamental concept in calculus used to find the area under a curve between two points. It is denoted by the integral symbol \(\int\) with limits of integration shown as the lower limit and upper limit. For instance, \(\int_{a}^{b} f(x) \, dx\) represents the area under the curve of the function \(f(x)\) from \(x = a\) to \(x = b\).
Characteristics of a Definite Integral:
  • It accumulates the net "signed" area, meaning areas above the x-axis are positive, and those below are negative.
  • The value of a definite integral is a single number, unlike an indefinite integral, which results in a family of functions.
  • It is essential for finding total values, such as total distance traveled, total area, or total volume.
In the given exercise, the definite integral \(\int_{1}^{x} f(t) \, dt = x^2 - 2x + 1\) represents the accumulated change from \(t=1\) to \(t=x\). Determining the function under this integral requires use of the Fundamental Theorem of Calculus, as seen in later sections.
Derivative
In calculus, the derivative of a function is a foundational tool used to determine the rate of change of a function concerning its variable. The derivative is symbolically represented as \(\frac{d}{dx}\), and it gives us the function's slope at any specific point.
Key Insights Into Derivatives:
  • The derivative of a constant is zero, as constants do not change.
  • The derivative tells you how the function value changes as the input changes.
  • Each function, like \(x^2\), has a specific rule for derivation, such as \(\frac{d}{dx}(x^2) = 2x\).
To solve the exercise, we differentiated the given expression \(x^2 - 2x + 1\). This operation relies on applying known rules to each term separately, leading to the result \(2x - 2\) which represents the original function \(f(x)\).
Differentiation
Differentiation is the action or process of computing a derivative. It's a critical technique in calculus used to find the instantaneous rate of change or slope of a curve. This process transforms a function into its derivative.
Steps in the Differentiation Process:
  • Identify the function to differentiate.
  • Apply differentiation rules to specific terms, such as power, product, or chain rules as needed.
  • Simplify the resulting expression.
During the step-by-step solution of the exercise, we utilized differentiation to uncover \(f(x)\). Applying the Fundamental Theorem of Calculus, we found \(f(x)\) by differentiating \(x^2 - 2x + 1\), verifying the role of \(d/dx\) in translating integral expressions into their derivative forms, highlighting \(f(t)\) as \(2x - 2\).

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Most popular questions from this chapter

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100.200\), and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=e^{-x^{2}} \quad \text { on } \quad[0,1]$$

Find \(\lim _{x \rightarrow \infty} \frac{1}{\sqrt{x}} \int_{1}^{x} \frac{d t}{\sqrt{t}}\).

Graph the function and find its average value over the given interval. $$h(x)=-|x| \quad \text { on } \quad \text { a. }[-1,0], \text { b. }[0,1], \text { and } c .[-1,1]$$

Suppose that $$\int_{0}^{1} f(x) d x=3$$ Find $$\int_{-1}^{0} f(x) d x$$ if a. \(f\) is odd, \(\quad\) b. \(f\) is even.

Express the solutions of the initial value problems in terms of integrals. $$\frac{d y}{d x}=\sqrt{1+x^{2}}, \quad y(1)=-2$$
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