Problem 3 - Single Variable Calculus

Question

Find the derivative of the function. $$ F(x)=\int_{0}^{x} \sqrt{3 t+5} d t $$

Step-by-Step Solution

Answer

The derivative of the function \(F(x) = \int_{0}^{x} \sqrt{3t + 5} dt\) is \(F'(x) = \sqrt{3x + 5}\).

Step 1: Understand the problem

We need to find the derivative of the function \(F(x) = \int_{0}^{x} \sqrt{3t + 5} dt\). By using the Fundamental theorem of Calculus, we can evaluate the derivative of this integral by evaluating the integrand at x.

Step 2: Apply the Fundamental theorem of Calculus

According to the Fundamental theorem of Calculus, the derivative of the function \(F(x) = \int_{0}^{x} \sqrt{3t + 5} dt\) with respect to x is equal to the value of the integrand evaluated at x. So, we can write the derivative as: \[F'(x) = \sqrt{3x + 5}\]

Step 3: Write the final answer

The derivative of the function \(F(x) = \int_{0}^{x} \sqrt{3t + 5} dt\) is: \[F'(x) = \sqrt{3x + 5}\]

Key Concepts

Understanding Derivatives

The concept of a derivative is fundamental in calculus. It represents the rate at which a function is changing at any given point. Think of it like a speedometer for functions—it tells us how quickly something is happening. When we take the derivative of a function, we are essentially looking at how the output changes as the input is varied minutely.

For example, if we have a function that describes the position of a car over time, the derivative of this function would tell us the car's velocity. Mathematically speaking, the derivative of a function \( f(x) \) is denoted by \( f'(x) \) or \( \frac{df}{dx} \), and it provides the slope of the tangent line to the curve at any point \( x \).

In the context of the given problem, finding the derivative involves applying the Fundamental Theorem of Calculus to evaluate the change in the integral function \( F(x) \).

Definite Integrals Explained

Definite integrals are a key concept when dealing with the area under a curve within a specified interval. Unlike indefinite integrals, which compute a general form of antiderivative, definite integrals give a specific numerical value that represents an area.

A definite integral is written as \( \int_{a}^{b} f(t) \, dt \), with \( a \) and \( b \) denoting the limits of integration. This tells us to find the total accumulation of the function \( f(t) \) from \( a \) to \( b \). The process involves calculating not just an antiderivative, but also the difference between the evaluations at these limits.

In our exercise, \( x \) serves as the upper limit of the integral, and the Fundamental Theorem of Calculus helps us transition from this integral to its derivative.

Integrand Evaluation and Its Role

Evaluating the integrand is crucial in understanding how variables are handled during integration and differentiation. The integrand is the function that appears under the integral sign, and in our exercise, it is \( \sqrt{3t + 5} \).

When using the Fundamental Theorem of Calculus to differentiate an integral where the upper limit is a variable \( x \), the derivative of the integral simply involves plugging \( x \) into the integrand. This makes the process much smoother and directly provides \( F'(x) \).

Thus, for \( F(x) = \int_{0}^{x} \sqrt{3t + 5} \, dt \), the derivative is derived by evaluating the integrand at \( t = x \). The result is \( F'(x) = \sqrt{3x + 5} \), elegantly linking the derivative with the originally integrated function.