Questions: If f(x)=2x^2-7x+7, find f'(2) Use this to find the equation of the tangent line to the parabola y=2x^2-7x+7 at the point (2,1). The equation of this tangent line can be written in the form y=mx+b where m is: and where b is:

If f(x)=2x^2-7x+7, find f'(2)
Use this to find the equation of the tangent line to the parabola y=2x^2-7x+7 at the point (2,1). The equation of this tangent line can be written in the form y=mx+b
where m is: and where b is:

Transcript text: If $f(x)=2 x^{2}-7 x+7$, find $f^{\prime}(2)$ $\square$ Use this to find the equation of the tangent line to the parabola $y=2 x^{2}-7 x+7$ at the point $(2,1)$. The equation of this tangent line can be written in the form $y=m x+b$ where $m$ is: $\square$ and where $b$ is: $\square$

Solution

Solution Steps

Step 1: Find the derivative of the quadratic function

The derivative of the function $f(x) = 2x^2 - 7x + 7$ is $f'(x) = 2ax + b$, which simplifies to $f'(x) = 4x - 7$.

Step 2: Evaluate the derivative at $x_0$ to find the slope $m$ of the tangent line

Substituting $x_0 = 2$ into $f'(x)$, we get $m = 4*2 - 7 = 1$.

Step 3: Find the y-coordinate of the point on the parabola

Substituting $x_0 = 2$ into the original function $f(x)$, we get $f(x_0) = 2_2^2 - 7_2 + 7 = 1$.

Step 4: Use the point-slope form of a line to find the equation of the tangent line

With $m = 1$ and the point $(2, 1)$, the equation of the tangent line is $y - 1 = 1(x - 2)$.

Step 5: Solve for $b$

Rearranging the equation to solve for $b$, we find $b = 1 - 1*2 = -1$.

Final Answer:

The equation of the tangent line to the parabola described by $f(x) = 2x^2 - 7x + 7$ at the point $(2, 1)$ is $y = x - 1$.

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