Questions: The slope of the tangent line at (x) is given by (f'(x)=7x^6+12x^5-6). Find (f(x)) if (f(-1)=18). (f(x)=)

Solution Steps

To find \( f(x) \), we need to integrate the derivative \( f'(x) = 7x^6 + 12x^5 - 6 \). After integrating, we'll use the initial condition \( f(-1) = 18 \) to solve for the constant of integration.

To find \( f(x) \), we integrate the given derivative:

\[ f'(x) = 7x^6 + 12x^5 - 6 \]

The integral of \( f'(x) \) is:

\[ f(x) = \int (7x^6 + 12x^5 - 6) \, dx = \frac{7}{7}x^7 + \frac{12}{6}x^6 - 6x + C = x^7 + 2x^6 - 6x + C \]

We use the initial condition \( f(-1) = 18 \) to find the constant \( C \):

\[ f(-1) = (-1)^7 + 2(-1)^6 - 6(-1) + C = 18 \]

Simplifying:

\[ -1 + 2 + 6 + C = 18 \]

\[ C + 7 = 18 \]

\[ C = 11 \]

Substitute \( C = 11 \) back into the expression for \( f(x) \):

\[ f(x) = x^7 + 2x^6 - 6x + 11 \]

Final Answer