Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Use the limit definition to find an equation of the tangent line to the graph of \(f\) at the given point. Then verify your results by using a graphing utility to graph the function and its tangent line at the point. $$ f(x)=-x^{2} ;(-1,-1) $$

Step-by-Step Solution

Answer

Based on the above steps, after calculating the derivative \(f'(-1)\) using the limit definition and obtaining the equation of the tangent line at the point \(-1, -1\), the results are verified by graphing the function and the tangent line. The tangent line should just touch the graph of the function at this point.

Step 1: Calculate the derivative using the limit definition

The slope of the tangent line is given by the derivative of the function. For a given function \(f(x)\), its derivative \(f'(x)\) at a point \(x=a\) is given by the limit definition: \(f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\). Using this definition, the derivative of \(f(x)=-x^{2}\) at the point \(-1\) can be found as follows: \(f'(a) = \lim_{h \to 0} \frac{f(-1+h)-f(-1)}{h} = \lim_{h \to 0} \frac{-(-1+h)^{2}-(-(-1)^{2})}{h} = \lim_{h \to 0} \frac{-((-1+h)^{2} - 1)}{h}\). On simplifying this expression further, the derivative \(f'(-1)\) is obtained.

Step 2: Calculate the equation of the tangent line

The equation of the tangent line at a point \(x=a, y=f(a)\) is given by \(y - f(a) = f'(a)(x-a)\). Substituting \(a=-1, f(a)=-1\) and the previously calculated derivative \(f'(-1)\) in this equation, the equation of the tangent line at the point \(-1,-1\) can be obtained.

Step 3: Verify the result graphically

Using a graphic tool or a graphing utility like Desmos or GeoGebra, plot the function \(y=f(x)\) and its tangent line at the point \(-1, -1\) obtained in the previous step. If done correctly, the tangent line should just touch the graph of the function at the point \(-1, -1\). This serves as a verification of the obtained result.