### Calculus Problem: Finding Derivatives and Tangent Lines#### Problem Statement:a. Find the derivative function f' for the function f.b. Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.#### Given Function and Value:\[ f(x) = 2x^2 - 9x + 10 \]\[ a = 2 \]#### Solutions:**a. Derivative Function f'**\[ f'(x) = 4x - 9 \]**b. Tangent Line Equation**To find the equation of the tangent line, we substitute \( a = 2 \) into the derivative function to find the slope at that point. Then, we find the y-coordinate by substituting \( x = 2 \) into the original function.The tangent line equation will be in the form:\[ y = mx + c \]Where \( m \) is the slope and \( c \) is the y-intercept.**Steps to Find the Tangent Line:**1. **Slope of the Tangent Line:** Substitute \( a = 2 \) into the derivative function to get the slope \( m \): \[ f'(2) = 4(2) - 9 = 8 - 9 = -1 \]2. **Point on the Graph:** Substitute \( x = 2 \) into the original function to get the y-coordinate \( f(2) \): \[ f(2) = 2(2)^2 - 9(2) + 10 = 2(4) - 18 + 10 = 8 - 18 + 10 = 0 \]3. **Equation of the Tangent Line:** Using the slope \( m \) and the point (2, 0): \[ y = -1(x - 2) + 0 \] Which simplifies to: \[ y = -x + 2 \]Thus, the equation of the tangent line is:\[ y = -x + 2 \]

Name: ### Calculus Problem: Finding Derivatives and Tangent Lines#### Probl
Uploaded: 2024-03-20T06:47:22.000Z
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Description: ### Calculus Problem: Finding Derivatives and Tangent Lines#### Problem Statement:a. Find the derivative function f' for the function f.b. Find an equation of the line tangent to the graph of f at (a, f(a)) for the given value of a.#### Given Functi