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Chapter 2: Problem 9

Find the slope of the tangent line to the graph of the function at the given point. \(f(t)=3 t-t^{2}, \quad(0,0)\)

Short Answer

Expert verified
The slope of the tangent line to the graph of the function \(f(t)=3t-t^{2}\) at the point (0,0) is 3.

Step by step solution

01

Find the derivative of the function

The derivative of a function gives us the slope of the function at any point. The derivative of \(f(t)=3t-t^{2}\) can be computed using the power rule for derivatives, which states that the derivative of \(t^{n}\) is \(n*t^{n-1}\). Therefore, the derivative of \(f\) is \[f'(t) = 3-2t\].
02

Evaluate the derivative at the given point

Now we evaluate the derivative at the x-coordinate of the given point to find the slope of the tangent line at that point. The x-coordinate of the given point is 0, so we find \(f'(0)\). Substituting 0 for \(t\) in the derivative gives us \[f'(0) = 3-2*0 = 3\]. Therefore, the slope of the tangent line to the graph of the function at the point (0,0) is 3.

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