Questions: Let f(x)=4x^4-x^2-3. (a) Find f'(x). f'(x)= (b) Find the slope of the line tangent to the graph of f at x=4. Slope at x=4 : (c) Find an equation of the line tangent to the graph of f at x=4. Tangent line: y=

Let f(x)=4x^4-x^2-3.
(a) Find f'(x).
f'(x)=

(b) Find the slope of the line tangent to the graph of f at x=4.

Slope at x=4 :

(c) Find an equation of the line tangent to the graph of f at x=4.

Tangent line: y=

Transcript text: Let $f(x)=4 x^{4}-x^{2}-3$. (a) Find $f^{\prime}(x)$. \[ f^{\prime}(x)= \] $\square$ (b) Find the slope of the line tangent to the graph of $f$ at $x=4$. Slope at $x=4$ : $\square$ (c) Find an equation of the line tangent to the graph of $f$ at $x=4$. Tangent line: $y=$ $\square$

Solution

Solution Steps

To solve the given problems, we need to follow these steps:

(a) To find the derivative $ f'(x) $, we will use the power rule for differentiation.

(b) To find the slope of the tangent line at $ x = 4 $, we will evaluate the derivative $ f'(x) $ at $ x = 4 $.

(c) To find the equation of the tangent line at $ x = 4 $, we will use the point-slope form of the equation of a line. We need the slope from part (b) and the value of the function $ f(x) $ at $ x = 4 $.

Step 1: Find the Derivative $ f'(x) $

Given the function $ f(x) = 4x^4 - x^2 - 3 $, we need to find its derivative. Using the power rule for differentiation, we get: \[ f'(x) = \frac{d}{dx}(4x^4) - \frac{d}{dx}(x^2) - \frac{d}{dx}(3) = 16x^3 - 2x \]

Step 2: Find the Slope of the Tangent Line at $ x = 4 $

To find the slope of the tangent line at $ x = 4 $, we evaluate the derivative $ f'(x) $ at $ x = 4 $: \[ f'(4) = 16(4)^3 - 2(4) = 16 \cdot 64 - 8 = 1024 - 8 = 1016 \]

Step 3: Find the Equation of the Tangent Line at $ x = 4 $

To find the equation of the tangent line, we use the point-slope form of the equation of a line. First, we need the value of the function $ f(x) $ at $ x = 4 $: \[ f(4) = 4(4)^4 - (4)^2 - 3 = 4 \cdot 256 - 16 - 3 = 1024 - 16 - 3 = 1005 \] Using the point-slope form $ y - y_1 = m(x - x_1) $, where $ m $ is the slope and $ (x_1, y_1) $ is the point $ (4, 1005) $: \[ y - 1005 = 1016(x - 4) \] Simplifying, we get: \[ y = 1016x - 4064 + 1005 = 1016x - 3059 \]