Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the tangent line to the graph of \(f\) at the point \((2, f(2)) .\) Use a graphing utility to check your result by graphing the original function and the tangent line in the same viewing window. $$ f(x)=\left(4-3 x^{2}\right)^{-2 / 3} $$

Step-by-Step Solution

Answer

With the calculations provided, firstly the derivative of the given function at the given point is calculated, from this the slope of the tangent is derived. Then using point-slope form of a line we construct the equation for the tangent line at the given point, (2, f(2)). Finally, we graph both the function and the derived tangent line on the same graphing utility, to verify that we got the calculations correct.

Step 1: Calculate the Derivative of the Function

We can find the derivative of the function \( f(x) \) using the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here's how to derive the derivative of \( f(x) =(4-3x^{2})^{-2 / 3} \): Firstly, find the derivative of the outer function, \( (u)^{-2 / 3} \). Letting \( u = 4 - 3x^2 \), the derivative of the outer function will be \( -2/3u^{-5/3} \), then multiply this by the derivative of the inner function \( u \), which is \( -6x \). This gives us the full derivative, \( f'(x) = 4x(4-3x^{2})^{-5 / 3} \).

Step 2: Evaluate the Derivative at the Given Point

Next, to find the slope of the tangent line at the point (2, f(2)), simply substitute the x-coordinate, 2, into the derivative equation: \( f'(2) = 4(2)(4-3*(2)^{2})^{-5 / 3} \). Simplifying this will give us the slope of the tangent line at the point (2, f(2)).

Step 3: Construct the Equation of the Tangent Line

Now with the slope of the tangent line and the point of tangency (2, f(2)), we can find the equation of the tangent line using the point-slope form of a line, \( y - y_1 = m*(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency. Substituting the slope from step 2 and the point (2, f(2)) will give us the equation of the tangent line.

Step 4: Confirm the Result by Graphing

Finally, draw both the original function \( f(x) = (4-3x^{2})^{-2 / 3} \) and the derived tangent line on the same graphing utility. If the tangent line just 'touches' the function curve at the point (2, f(2)), we have done everything correctly.