Find the Antiderivative f(t)=0.8t-15.9

Solution:

Step 1:

Identify the antiderivative $F(t)$ by integrating the given function $f(t)$.

$F(t) = \int f(t) \, dt$

Step 2:

Write down the integral that needs to be solved.

$F(t) = \int (0.8t - 15.9) \, dt$

Step 3:

Decompose the integral into separate terms.

$\int 0.8t \, dt + \int (-15.9) \, dt$

Step 4:

Extract the constant coefficient $0.8$ from the first integral.

$0.8 \int t \, dt - \int 15.9 \, dt$

Step 5:

Utilize the Power Rule for integration on the variable $t$.

$0.8 \left( \frac{t^2}{2} + C \right) - \int 15.9 \, dt$

Step 6:

Apply the rule for integrating constants.

$0.8 \left( \frac{t^2}{2} + C \right) - 15.9t + C$

Step 7:

Simplify the expression.

Step 7.1:

Combine the constants with the variable term.

$0.8 \left( \frac{t^2}{2} + C \right) - 15.9t + C$

Step 7.2:

Perform the simplification.

$\frac{4}{5} \cdot \frac{t^2}{2} - 15.9t + C$

Step 7.3:

Reorder the terms for the final antiderivative.

$\frac{4t^2}{10} - 15.9t + C$

Step 8:

Present the antiderivative of the function $f(t) = 0.8t - 15.9$.

$F(t) = \frac{4t^2}{10} - 15.9t + C$

Knowledge Notes:

The process of finding an antiderivative, also known as the indefinite integral, involves reversing the process of differentiation. Here are the relevant knowledge points used in solving the problem:

1. Indefinite Integral: The antiderivative of a function $f(t)$ is represented by the integral sign without bounds, $\int f(t) \, dt$, and includes an arbitrary constant $C$ since the derivative of a constant is zero.

2. Linearity of Integration: The integral of a sum of functions is equal to the sum of their integrals. This property allows us to split the integral of $0.8t - 15.9$ into two separate integrals.

3. Constant Multiple Rule: If a constant is multiplied by a function, it can be factored out of the integral. This is why we can move $0.8$ outside of the integral in Step 4.

4. Power Rule for Integration: For any real number $n \neq -1$, the integral of $t^n$ with respect to $t$ is $\frac{t^{n+1}}{n+1} + C$. This is applied in Step 5 where $n=1$.

5. Constant Rule: The integral of a constant is equal to the constant multiplied by the variable of integration. This is applied in Step 6 for the term $-15.9$.

6. Simplification: Combining like terms and simplifying coefficients is a standard algebraic process applied in Step 7 to obtain the final antiderivative expression.

In summary, the problem-solving process involves recognizing the structure of the function to be integrated, applying the rules of integration, and simplifying the result to find the antiderivative.