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Chapter 1: Problem 148

In the following exercises, use the Fundamental Theorem of Calculus, Part 1 , to find each derivative. $$ \frac{d}{d x} \int_{1}^{x} e^{-t^{2}} d t $$

Short Answer

Expert verified
The derivative is \( e^{-x^{2}} \).

Step by step solution

01

Identify the Function and Interval

The problem is to find the derivative of the function defined by the integral \( \int_{1}^{x} e^{-t^{2}} \, dt \). This integral represents a function where the upper limit of integration is the variable \( x \).
02

Apply the Fundamental Theorem of Calculus, Part 1

The Fundamental Theorem of Calculus, Part 1 states that if \( F(x) = \int_{a}^{x} f(t) \, dt \), then the derivative \( \frac{d}{dx} F(x) = f(x) \). Here, \( f(t) = e^{-t^{2}} \) and the integral is from 1 to \( x \).
03

Differentiate Using the Theorem

By applying the theorem, we conclude that the derivative of the integral is simply the function evaluated at \( x \). Thus, \( \frac{d}{d x} \int_{1}^{x} e^{-t^{2}} \, dt = e^{-x^{2}} \).

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Key Concepts

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

Derivative
A derivative is a fundamental tool in calculus and is essential for understanding how functions change. In simplest terms, a derivative measures the rate at which a quantity changes with respect to a change in another quantity. For example, it can describe how fast a car is moving at a particular moment.
In mathematical terms, suppose we have a function denoted as \( f(x) \). The derivative, written as \( f'(x) \), represents the function鈥檚 rate of change concerning the variable \( x \). The notation \( \frac{d}{dx} \) is also commonly used to denote derivatives.
  • The derivative of a function \( f(x) \) at a point \( x \) represents the slope of the tangent line to the function's graph at that point.
  • Derivatives can be used for finding minima and maxima of functions which is crucial in optimization problems.
  • In this exercise, we are asked to find the derivative of an integral, which involves the Fundamental Theorem of Calculus.
Definite Integral
A definite integral, encountered frequently in calculus, is a way to find the total accumulated value, taking into account the process over an interval. Unlike indefinite integrals, which represent a family of functions, a definite integral gives a precise numerical result.
In this exercise, the integral is from a fixed lower limit (1) to a variable upper limit (\( x \)). The function inside the integral is \( e^{-t^{2}} \), a common expression related to Gaussian functions.
  • Definite integrals can be thought of as the area under a curve within a specified range.
  • They are fundamentally connected to the derivative, as highlighted by the Fundamental Theorem of Calculus.
  • In our case, the integral from 1 to \( x \) serves as a foundation to apply this theorem.
Calculus Exercises
Practicing calculus problems is crucial to mastering the concepts like derivatives and integrals. Each exercise offers a chance to apply theoretical knowledge to practical problems, reinforcing understanding and skills.
This particular exercise utilizes the Fundamental Theorem of Calculus to find a derivative, demonstrating the interconnected nature of calculus concepts.
  • Applying the Fundamental Theorem allows you to easily differentiate a function represented by a definite integral.
  • Effective practice involves approaching problems step-by-step, as seen in the original solution, to deepen your understanding of the methods involved.
  • Frequent practice enables students to recognize patterns and apply calculus principles confidently across varied problems.

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Most popular questions from this chapter

Pretend, for the moment, that we do not know that \(e^{x}\) is the inverse function of \(\ln x,\) but keep in mind that \(\ln x\) has an inverse function defined on \((-\infty, \infty) .\) Call it \(E\). Show that \(E^{\prime}(t)=E(t)\).

Find the area under the graph of the function \(f(x)=x e^{-x^{2}}\) between \(x=0\) and \(x=5\)

In the following exercises, \(f(x) \geq 0\) for \(a \leq x \leq b\). Find the area under the graph of \(f(x)\) between the given values \(a\) and \(b\) by integrating. $$ f(x)=\frac{\log _{2}(x)}{x} ; a=32, b=64 $$

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{1} \frac{x}{\sqrt{1+x^{2}}} d x $$

[T] The graph below plots the cubic \(p(t)=0.07 t^{3}+2.42 t^{2}-25.63 t+521.23\) against the data in the preceding table, normalized so that \(t=0\) corresponds to \(1963 .\) Estimate the average number of bald eagles per year present for the 37 years by computing the average value of \(p\) over [0,37] .
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