Problem 27 - Calculus Single Variable

Question

In each of Exercises \(21-28,\) calculate the derivative of \(F(x)\) with respect to \(x\). $$ F(x)=\int_{-1}^{x} t \sqrt{1}+t d t $$

Step-by-Step Solution

Answer

The derivative is \( F'(x) = x \sqrt{1+x} \).

Step 1: Identify the Integral Form

The given function is \( F(x) = \int_{-1}^{x} t \sqrt{1}+t \, dt \). This is an integral with a variable upper limit \( x \).

Step 2: Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \( F(x) = \int_{a}^{x} f(t) \, dt \), then the derivative of \( F(x) \) with respect to \( x \) is \( F'(x) = f(x) \).

Step 3: Identify the Integrand

In this exercise, the integrand \( f(t) \) is \( t \sqrt{1}+t = t \cdot (1+t)^{1/2} \) (considering \( \sqrt{1} + t = \sqrt{1+t} \), assuming correctly interpreted as per context).

Step 4: Differentiate Using the Theorem

By directly applying the Fundamental Theorem of Calculus, we replace \( t \) with \( x \), hence \( F'(x) = x \sqrt{1}+x = x \cdot (1+x)^{1/2} \).

Key Concepts

Derivative

Understanding derivatives is crucial in calculus as they help you determine the rate at which a quantity changes. In essence, the derivative measures the steepness or slope of a curve at any given point. To calculate the derivative, one takes the limit of the average rate of change of the function as the interval approaches zero.
You can think of it as zooming in so closely on the curve that it begins to look like a straight line, and the slope of this line is your derivative.

• The general notation for a derivative is \( f'(x) \) or \( \frac{df}{dx} \).
• For simple functions, there are rules such as the power rule, product rule, and chain rule that simplify finding derivatives.
• Derivatives are used in many applications such as finding the velocity in physics, optimizing systems, or dealing with rates of change in economics.

Knowing how to differentiate is not just about mathematics; it's about understanding how and why things change, which is a valuable skill in any field.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus serves as a vital link between differentiation and integration. It comprises two parts that provide deep insights into the nature of calculus:

1. **First Part**: This part states that if you have a function \( f \) that is continuous over an interval \([a, b]\), and you define a function \( F \) as the integral of \( f \) from \( a \) to \( x \), then \( F \) is continuous over \([a, b]\) and differentiable over \((a, b)\), with \( F'(x) = f(x) \).
This makes it possible to find the derivative of the integral function equal to the original function used in the integration.
2. **Second Part**: It helps us compute definite integrals. If \( F \) is an antiderivative of \( f \) on \([a, b]\), then: \[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]The theorem elegantly bridges the concepts of area under a curve (integrals) and the rate of change or slope (derivatives), revealing that they are inverse processes.
Without it, calculus wouldn’t have the powerful applications we leverage today across sciences and engineering.

Integrals

Integrals are a cornerstone of calculus, providing a way to accumulate quantities and calculate areas under curves.
When you integrate a function, you essentially add up infinitely small quantities to find the total change or area.
Integrals come in two main types:

• **Indefinite Integrals**: Represented as \( \int f(x) \, dx \), an indefinite integral gives the antiderivative of \( f(x) \). Think of it as reversing the process of differentiation.
  
  • The result always contains a constant \( C \), since differentiation of any constant is zero, thus it gets lost during differentiation.
  • Example: \( \int x^2 \, dx = \frac{x^3}{3} + C \).
• **Definite Integrals**: Represented as \( \int_{a}^{b} f(x) \, dx \), a definite integral calculates the net area between the curve, \( f(x) \), and the x-axis over a specific interval \([a, b]\).
  
  • This type provides a numerical value unlike indefinite integrals.
  • Used in a wide range of applications where total accumulated value is important, such as finding distances, areas, and in probability theory.

Integrals allow us to deal with the accumulation and distribution of quantities, giving us tools to solve real-world problems where summing a series of infinitely small changes is needed, maintaining their relevance in both academic and real-world applications.