Problem 2 - Applied Calculus

Question

Find an antiderivative. $$ f(t)=5 t $$

Step-by-Step Solution

Answer

The antiderivative is \( F(t) = \frac{5t^2}{2} + C \).

Step 1: Understanding the Problem

We need to find an antiderivative of the function \( f(t) = 5t \). An antiderivative (or indefinite integral) of a function is a function whose derivative is the original function. This means we need to find a function \( F(t) \) such that \( F'(t) = 5t \).

Step 2: Identifying the Formula

Recall that the integral of \( t^n \) is given by \( \frac{t^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. In this case, \( f(t) = 5t \) can be rewritten as \( 5t^1 \).

Step 3: Applying the Integral Formula

Using the formula, we find the antiderivative of \( f(t) = 5t \) by integrating: \[ \int 5t \, dt = 5 \int t^1 \, dt = 5 \left( \frac{t^{1+1}}{1+1} \right) + C = 5 \left( \frac{t^2}{2} \right) + C. \]

Step 4: Simplifying the Expression

Simplify the expression obtained from the integration step: \[ \frac{5t^2}{2} + C. \] This is the antiderivative of \( f(t) = 5t \).

Key Concepts

Indefinite Integral

When you hear the term "indefinite integral," think of it as the reverse of taking a derivative. While a derivative tells you the rate at which a function is changing, an indefinite integral, or antiderivative, tells you the original function before it was differentiated.
An indefinite integral is represented by the integral symbol followed by a function and "dx," like this: \( \int f(x) \, dx \). Here’s what you need to understand:

• The indefinite integral includes a family of functions, not just one.
• This family differs by a constant since when you differentiate a constant, you get zero.
• The process of finding an indefinite integral is also called integration. It's like performing reverse differentiation.

When solving the problem \( f(t) = 5t \), we found its indefinite integral by applying integration rules that reverse the differentiation process.

Integration Formula

The integration formula is fundamental to solving integrals. If you know the right formula, finding the antiderivative becomes way easier. In this exercise, the key integration formula used is:

• \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \)
  This formula helps find the antiderivative of any function of the form \( t^n \). The symbol \( C \) stands for the constant of integration, which we'll discuss next.
• For the function \( f(t) = 5t \), we treat it as \( 5t^1 \).
• Apply the formula: First, increase the exponent by 1 (so, 1 becomes 2). Then, divide by this new exponent.

Using our integration formula, the antiderivative of \( f(t) = 5t \) becomes \( \frac{5t^2}{2} + C \). This formula is extremely useful for any polynomial function, so keep it close!

Constant of Integration

The constant of integration is a special part of finding indefinite integrals. Whenever you integrate a function without limits, you must add this constant, represented as \( C \), to your solution.

Here's why it's important:

• When differentiating, constants "disappear" because their derivatives are zero.
• This means there could have been any constant present before we differentiated the original function.
• Adding \( C \) ensures we cover all possible original functions with all sorts of constants.

In our problem of finding the antiderivative \( \int 5t \, dt \), we included \( C \) to acknowledge that there are infinitely many solutions. Each different \( C \) gives a different antiderivative, which are all correct.