Question

Lines tangent to parabolas a. Find the derivative function \(f^{\prime}\) for the following functions \(f\) b. Find an equation of the line tangent to the graph of \(f\) at \((a, f(a))\) for the given value of \(a\) c. Graph \(f\) and the tangent line. $$f(x)=3 x^{2} ; a=0$$

Short answer

Answer: The equation of the tangent line is \(y = 0\).

Step-by-step solution

Find the derivative function \(f^{\prime}\)

To find the derivative of a function, we need to find the rate of change of the function with respect to the independent variable \(x\). In our case, the given function is \(f(x) = 3x^2\). Applying the power rule of differentiation, we get: $$f^{\prime}(x)=\frac{d}{dx}(3x^{2})=6x$$

Evaluate the derivative at the given point \(a = 0\)

Now, to find the slope of the tangent line at \((a, f(a))\), we need to evaluate the derivative \(f^{\prime}(x)\) at the point \(a = 0\): $$f^{\prime}(a)=f^{\prime}(0)=6(0)=0$$ This means that the slope of the tangent line at the point \((a, f(a)) = (0, f(0))\) is \(0\).

Calculate \(f(a)\)

To find the coordinates of the point at which the tangent line touches the curve, we need to evaluate the function \(f(x)\) at \(a = 0\): $$f(a)=f(0)=3(0)^2=0$$ So, the point of tangency is \((0,0)\).

Write the equation of the tangent line

Since we have found the slope \(m\) of the tangent line to be \(0\) and the point of tangency \((a, f(a)) = (0,0)\), we can use the point-slope form of a linear equation to write the equation of the tangent line: $$y - f(a) = m(x - a)$$ Plugging in the values, we get: $$y - 0 = 0(x - 0)$$ $$y = 0$$ This is the equation of the tangent line to the graph of \(f(x) = 3x^2\) at \((0,0)\).

Graph the function and the tangent line

Now that we have both the equation of the function \(f(x) = 3x^2\) and the equation of the tangent line \(y = 0\), we can graph them together. On the graph, you will see a parabola representing the function \(f(x)\) and a horizontal line representing the tangent line \(y=0\) at the point \((0,0)\).