Question

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{7}^{x} \frac{1+t}{1+t^{2}} d t$$

Short answer

The derivative, \( dy/dx \), is \( \frac{1+x}{1+x^{2}} \)

Step-by-step solution

Identify the Function in the Integral

The function in the integral is \( f(t) = \frac{1+t}{1+t^{2}} \). This function is continuous over all real numbers.

Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, the derivative of \( \int_{7}^{x} f(t) dt \) is simply \( f(x) \). Hence, \( y' = \frac{1+x}{1+x^{2}} \).

Key concepts

Derivatives

Understanding derivatives is essential when it comes to applying the Fundamental Theorem of Calculus. A derivative represents the rate at which a function is changing at any given point. It is the slope of the tangent to the function's curve at any point. Essentially, derivatives describe how a quantity varies with respect to a change in another quantity, typically time.

In the given exercise, we find the derivative of an integral function. Thanks to the power of the Fundamental Theorem of Calculus, we know that the derivative of the integral from a constant to a variable upper limit is simply the original function evaluated at the variable limit.

Understanding this concept helps simplify many problems in calculus by transforming them into basic differentiation problems. The key takeaway is to recognize that, when differentiated, \( \int_{7}^{x} f(t) dt \) becomes \( f(x) \) when \( f(t) \) is a continuous function over the relevant interval.

Indefinite Integrals

Let's also briefly cover the concept of indefinite integrals. An indefinite integral, symbolically represented by \( \int f(x)dx \), is essentially the family of all antiderivatives of the function \( f(x) \). Rather than calculating the area under a curve from one point to another, indefinite integrals reverse the process of differentiation. That is, they help us find functions when we only know their derivatives.

In the context of our exercise, we are dealing with a definite integral from 7 to \( x \). However, the process of finding the antiderivative is similar to that of an indefinite integral, save for the bounds. If we had not been given limits of integration, we could express the antiderivative as \( F(x) + C \), where \( C \) is the constant of integration to account for all potential antiderivatives.

Indefinite integrals play a crucial role in solving problems involving the accumulation of quantities and the reconstruction of functions from their rates of change.

Continuous Functions

The significance of continuous functions in the realm of calculus cannot be overstated. A function is continuous at a point if there is no interruption in the graph of the function at that point; the function is defined there, and it smoothly connects to its surroundings in the domain.

In our example, \( f(t) = \frac{1+t}{1+t^{2}} \) is continuous over all real numbers. Continuity is paramount when applying the Fundamental Theorem of Calculus because it ensures that there are no points of discontinuity that could complicate the process of taking derivatives or evaluating integrals within a range.

The continuity of \( f(t) \) ensures that we can safely apply the Fundamental Theorem of Calculus and expect that the derivative of the integral will match the original function, provided the integral has a variable upper limit. Thus, grasping the behavior of continuous functions enhances the understanding of how operations like differentiation and integration apply to real-world, smoothly varying phenomena.