Question

Find the slope of the graph of the function at the indicated point. Use the derivative feature of a graphing utility to confirm your results. $$ f(x)=-\frac{1}{2}+\frac{7}{5} x^{3} $$ $$ \left(0,-\frac{1}{2}\right) $$

Short answer

The slope of the graph of the function \( f(x)=-\frac{1}{2}+\frac{7}{5} x^{3} \) at the point \( (0,-\frac{1}{2}) \) is 0.

Step-by-step solution

Find the Derivative

Let's start by finding the derivative of the function. We will use the power rule, which states that the derivative of \(x^n\) is \(n*x^{n-1}\). Hence, the derivative of \(f(x)= -\frac{1}{2}+\frac{7}{5}x^3\) is \(f'(x) = \frac{21}{5}x^2\).

Evaluate the Derivative at the Point

Now, we'll plug in the given x-value into the derivative. Here, x=0. So, \(f'(0) = \frac{21}{5}*0^2 = 0\).

Interpret the Result

Interpret the answer. The point of interest was (0,-1/2) and the derivative at this point is 0, meaning at this point the slope of the tangent line to the function is 0.