Question

How to find $(f)$ if $f'(x) = 16{x^3} + 6x + 7$ and $f( - 1) = - 1$?

Answer 1

Hint: Substitute the values in place of the terms to make the equation easier to solve. Then we will differentiate the term. Now we will substitute these terms in the original expression and integrate. Evaluate the integrals of both the sides separately. Integrate the terms separately.

Complete step by step solution:
The integration of any derived term is the term itself. So, here we need to evaluate the value of the function $(f)$ hence, we will integrate the derivative of $(f)$ which is $f'(x)$.
First, we will start off by integrating both the sides separately.
We will start by the integration of the left hand side.
$
= \int {f'(x)dx} \\
= f(x) \\
$
Now we will integrate the right hand side of the expression which is $16{x^3} + 6x + 7$.
\[
= \int {16{x^3} + 6x + 7} dx \\
= \int {16{x^3}dx + \int {6xdx} + \int {7dx} } \\
= \dfrac{{16{x^4}}}{4} + \dfrac{{6{x^2}}}{2} + 7x + c \\
= 4{x^4} + 3{x^2} + 7x + c \\
\]
Hence, the value of the function $(f)$ will be \[4{x^4} + 3{x^2} + 7x + c\].
Now to evaluate the value of the constant, we will substitute the value of $x$ as $ - 1$.
\[
\,\,f(x) = 4{x^4} + 3{x^2} + 7x + c \\
f( - 1) = 4{( - 1)^4} + 3{( - 1)^2} + 7( - 1) + c \\
\,\,\,\,\,\,\, - 1 = 4(1) + 3(1) - 7 + c \\
\,\,\,\,\,\, - 1 = 4 + 3 - 7 + c \\
\,\,\,\,\,\,\,\,c = - 1 \\
\]
Therefore, the value of the function $(f)$ will be \[4{x^4} + 3{x^2} + 7x - 1\].

Additional Information: A derivative is the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations. In general, scientists observe changing systems to obtain rate of change of some variable of interest, incorporate this information into some differential equation, and use integration techniques to obtain a function that can be used to predict the behaviour of the original system under diverse conditions.

Note: While substituting the terms make sure you are taking into account the degrees and signs of the terms as well. While applying the power rule make sure you have considered the power with their respective signs. Remember the integration of the terms.