Questions: For the function f(x)=7x^2-3x+5x^3, Step 2 of 2: Find the equation of the tangent line at x=4.

For the function f(x)=7x^2-3x+5x^3,
Step 2 of 2: Find the equation of the tangent line at x=4.
Transcript text: For the function $f(x)=7 x^{2}-3 x+5 x^{3}$, Step 2 of 2: Find the equation of the tangent line at $x=4$.
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Solution

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Solution Steps

Step 1: Find the Derivative

The function is given by \( f(x) = 5x^3 + 7x^2 - 3x \). To find the slope of the tangent line, we first calculate the derivative: \[ f'(x) = 15x^2 + 14x - 3 \]

Step 2: Evaluate the Derivative at \( x = 4 \)

Next, we evaluate the derivative at \( x = 4 \) to find the slope of the tangent line: \[ f'(4) = 15(4^2) + 14(4) - 3 = 293 \]

Step 3: Calculate the Function Value at \( x = 4 \)

We then calculate the value of the function at \( x = 4 \): \[ f(4) = 5(4^3) + 7(4^2) - 3(4) = 420 \]

Step 4: Write the Equation of the Tangent Line

Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m = 293 \), \( x_1 = 4 \), and \( y_1 = 420 \), we can express the equation of the tangent line: \[ y - 420 = 293(x - 4) \] Simplifying this gives: \[ y = 293x - 752 \]

Final Answer

The equation of the tangent line at \( x = 4 \) is \[ \boxed{y = 293x - 752} \]

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