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Chapter 6: Problem 14

Find \(\frac{d y}{d x} .\) $$ y=\int_{\pi / 4}^{x} \cos ^{2}(u-3) d u $$

Short Answer

Expert verified
The derivative is \(\cos^2(x-3)\).

Step by step solution

01

Identify the type of function

We need to differentiate a function defined by a definite integral with a variable upper bound, which is a classic application of the Fundamental Theorem of Calculus.
02

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if a function is defined as an integral with respect to a variable upper bound, the derivative of the function is the integrand evaluated at that upper bound. Specifically, if \(F(x) = \int_{a}^{x} f(u) \, du\), then \(\frac{d F}{d x} = f(x)\). In this problem, the integrand is \(\cos^2(u-3)\).
03

Differentiate the function

According to the theorem, we take the integrand \(\cos^2(u-3)\) and replace \(u\) with \(x\). Therefore, the derivative \(\frac{d y}{d x} = \cos^2(x-3)\).

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Key Concepts

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

Differentiation
Differentiation is a fundamental concept in calculus. It involves finding the rate at which a quantity changes. This is most often called the derivative. In practical terms, differentiation helps us understand how a function behaves at any point.Key points about differentiation include:
  • The process involves finding the derivative, indicated by symbols like \( rac{d}{dx} \) or \( f'(x) \).
  • The derivative tells us the slope of the tangent line to the function at any given point.
  • For a function \(y = f(x)\), the derivative \( rac{dy}{dx} \) represents the immediate rate of change of \(y\) with respect to \(x\).
In this exercise, we are differentiating a function that is initially defined in terms of a definite integral. Differentiating such functions often utilizes the power of the Fundamental Theorem of Calculus, which simplifies the process.
Definite Integrals
A definite integral is used to calculate the area under the curve of a function between two points. Unlike indefinite integrals, definite integrals have specific limits of integration. They result in a numerical value rather than a general expression.Characteristics of definite integrals:
  • In a definite integral, \( \int_a^b f(u) \, du \), \(a\) is the lower limit, and \(b\) is the upper limit.
  • It computes the net area under the function \(f(u)\) between \(a\) and \(b\).
  • Definite integrals are used in various applications, including calculating areas, volumes, and other quantities.
In the given exercise, the function \( y = \int_{\pi/4}^{x} \cos^2(u-3) \, du \) represents a definite integral with a variable upper bound, which is crucial for differentiating through the Fundamental Theorem of Calculus.
Variable Upper Bound
The concept of a variable upper bound in an integral is essential for applying the Fundamental Theorem of Calculus. This occurs when the upper limit of the integral is a variable, rather than a fixed number.The implications of having a variable upper bound:
  • A variable upper bound means the upper limit of integration changes depending on another variable (commonly \(x\)). This makes it a function of that variable.
  • For \( \int_a^{x} f(u) \, du \), differentiating with respect to \(x\) using the Fundamental Theorem of Calculus results in the integrand evaluated at \(x\). The derivative is \( f(x) \).
In our exercise, \( y = \int_{\pi/4}^{x} \cos^2(u-3) \, du \), changing the upper limit from a constant to a variable makes the function fit perfectly into the context of the Fundamental Theorem. It makes it straightforward to find the derivative \( \frac{dy}{dx} = \cos^2(x-3) \).

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Most popular questions from this chapter

Evaluate the definite integrals. $$ \int_{1}^{e} \frac{1}{x} d x $$

Compute the indefinite integrals. $$ \int(2 x+3)^{2} d x $$

Compute the indefinite integrals. $$ \int\left(x^{7 / 2}+x^{2 / 7}\right) d x $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{x}^{5} \frac{1}{u^{2}} d u, x>0 $$

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{2+x^{2}}^{2} \cot t d t $$
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