Problem 8 - Calculus: Early Transcendentals

Question

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \( \displaystyle g(x) = \int^x_1 \ln (1 + t^2) \,dt \)

Step-by-Step Solution

Answer

The derivative is \( g'(x) = \ln(1 + x^2) \).

Step 1: Understand the Fundamental Theorem of Calculus, Part 1

Part 1 of the Fundamental Theorem of Calculus states that if \( F(x) \) is an antiderivative of \( f(x) \) on an interval \( I \), then:\[ \frac{d}{dx} \int_{a}^{x} f(t) \, dt = f(x) \]for every \( x \) in \( I \). This means that if a function is defined as an integral with its upper limit as \( x \), then its derivative is simply the integrand evaluated at \( x \).

Step 2: Identify the Integrand

The given function is \( g(x) = \int^x_1 \ln (1 + t^2) \,dt \). Here, the integrand \( f(t) = \ln(1 + t^2) \).

Step 3: Apply the Fundamental Theorem of Calculus, Part 1

According to Part 1 of the Fundamental Theorem of Calculus, the derivative of \( g(x) = \int^x_1 \ln (1 + t^2) \, dt \) is the integrand evaluated at \( x \). So, \[ g'(x) = \ln(1 + x^2) \].

Step 4: Conclusion

The derivative of the function \( g(x) = \int^x_1 \ln (1 + t^2) \,dt \) is \( g'(x) = \ln(1 + x^2) \).

Key Concepts

Antiderivative

An antiderivative of a function is another function for which the original function is the derivative. Think of it as the reverse process of differentiation. If you have a function, say \( f(x) \), and there exists another function \( F(x) \) such that the derivative of \( F(x) \) is equal to \( f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \). It's important to note:

• An antiderivative is not unique; it can differ by a constant since the derivative of a constant is zero.
• Finding an antiderivative often involves reversing the operations performed during differentiation.

This concept is crucial in the Fundamental Theorem of Calculus, as it connects differentiation with integration. Understanding antiderivatives helps us solve problems involving areas under curves and accumulate quantities over intervals.

Integrand

The integrand is the function that you are integrating in an integral. When you see an integral like \( \int f(t) \, dt \), \( f(t) \) represents the integrand. The process of integration is essentially finding the antiderivative of this function.

• The integrand dictates the behavior of the integral; changing it alters the integral's value.
• Identifying the correct integrand is a critical step in solving problems using the Fundamental Theorem of Calculus.
• In our example, the integrand is \( \ln(1 + t^2) \), and applying the theorem allows us to evaluate this function at the upper limit to find the derivative.

Properly understanding what the integrand is and how it functions helps in applying various integration techniques effectively.

Definite Integral

A definite integral is an integral that computes the accumulation of quantities, such as area, between two bounds or limits. Unlike indefinite integrals, which represent a family of functions, definite integrals yield a specific numerical value. The notation \( \int_{a}^{b} f(t) \, dt \) indicates that we are finding the integral of the function \( f(t) \) from \( a \) to \( b \).

• The definite integral finds comprehensive application in calculating areas under curves, physical quantities such as mass and charge, and probabilities.
• Part 1 of the Fundamental Theorem of Calculus tells us how to compute the derivative of a function defined as a definite integral with an upper limit variable.
• The theorem reassures us that for a continuous function, this derivative is simply the integrand evaluated at the upper bound \( x \).

By using definite integrals, we can solve real-world problems that involve calculating the total change over an interval.