Questions: Now let's calculate the tangent line to the function f(x)=1-5 x^2 at x=4. a. By using f'(x) from part 2 , the slope of the tangent line to f at x=4 is f'(4)=.

Transcript text: Now let's calculate the tangent line to the function $f(x)=1-5 x^{2}$ at $x=4$. a. By using $f^{\prime}(x)$ from part 2 , the slope of the tangent line to $f$ at $x=4$ is $f^{\prime}(4)=$ $\square$

Solution

Solution Steps

To find the slope of the tangent line to the function $ f(x) = 1 - 5x^2 $ at $ x = 4 $, we need to calculate the derivative of the function, $ f'(x) $, and then evaluate it at $ x = 4 $.

Step 1: Find the Derivative of the Function

To find the slope of the tangent line to the function $ f(x) = 1 - 5x^2 $ at $ x = 4 $, we first need to calculate the derivative of the function. The derivative, $ f'(x) $, is given by:

\[ f'(x) = \frac{d}{dx}(1 - 5x^2) = -10x \]

Step 2: Evaluate the Derivative at $ x = 4 $

Next, we evaluate the derivative at $ x = 4 $ to find the slope of the tangent line at this point:

\[ f'(4) = -10 \times 4 = -40 \]