The function $f$ is defined by $f(x)=x^{2}-2 x+3 \cos \left(x^{2}-x\right)$ . Use a calculator to write the equation of the line tangent to the graph of $f$ when $x=-2.5$ . You should round all decimals to $3$ places. $\newline$ Answer:

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Find equations of tangent lines using limits

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Q. The function

f

is defined by

f(x)=x^{2}-2 x+3 \cos \left(x^{2}-x\right)

. Use a calculator to write the equation of the line tangent to the graph of

f

when

x=-2.5

. You should round all decimals to

3

places.

\newline

Answer:

Calculate Derivative: To find the equation of the tangent line to the graph of the function at $x = -2.5$ , we need to calculate the derivative of the function, which will give us the slope of the tangent line at that point.
Evaluate Derivative at $x = -2.5$ : The derivative of $f(x) = x^2 - 2x + 3\cos(x^2 - x)$ is $f'(x) = 2x - 2 - 3\sin(x^2 - x)(2x - 1)$ by using the chain rule and the product rule for differentiation.
Find Slope of Tangent Line: Now we need to evaluate the derivative at $x = -2.5$ to find the slope of the tangent line. So we calculate $f'(-2.5) = 2(-2.5) - 2 - 3\sin((-2.5)^2 - (-2.5))(2(-2.5) - 1)$ .
Calculate y-coordinate at $x = -2.5$ : Using a calculator, we find $f'(-2.5) = -5 - 2 - 3\sin(6.25 + 2.5)(-5 - 1)$ . We need to round all decimals to three places.
Find Point on Function: After calculating the value, we find $f'(-2.5) \approx -7 - 3\sin(8.75)(-6) \approx -7 - 3(-0.626)(-6) \approx -7 - 11.268 \approx -18.268$ (rounded to three decimal places).
Write Tangent Line Equation: Now we have the slope of the tangent line, which is $-18.268$ . Next, we need to find the y-coordinate of the point on the function where $x = -2.5$ . We calculate $f(-2.5) = (-2.5)^2 - 2(-2.5) + 3\cos((-2.5)^2 - (-2.5))$ .
Convert to Slope-Intercept Form: Using a calculator, we find $f(-2.5) \approx 6.25 + 5 + 3\cos(6.25 + 2.5) \approx 11.25 + 3\cos(8.75) \approx 11.25 + 3(0.684) \approx 11.25 + 2.052 \approx 13.302$ (rounded to three decimal places).
Convert to Slope-Intercept Form: Using a calculator, we find $f(-2.5) \approx 6.25 + 5 + 3\cos(6.25 + 2.5) \approx 11.25 + 3\cos(8.75) \approx 11.25 + 3(0.684) \approx 11.25 + 2.052 \approx 13.302$ (rounded to three decimal places).We now have the point $(-2.5, 13.302)$ and the slope $-18.268$ . The equation of the tangent line in point-slope form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the point on the line.
Convert to Slope-Intercept Form: Using a calculator, we find $f(-2.5) \approx 6.25 + 5 + 3\cos(6.25 + 2.5) \approx 11.25 + 3\cos(8.75) \approx 11.25 + 3(0.684) \approx 11.25 + 2.052 \approx 13.302$ (rounded to three decimal places).We now have the point $(-2.5, 13.302)$ and the slope $-18.268$ . The equation of the tangent line in point-slope form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the point on the line.Substituting the values into the point-slope form, we get $y - 13.302 = -18.268(x - (-2.5))$ or $y - 13.302 = -18.268(x + 2.5)$ .
Convert to Slope-Intercept Form: Using a calculator, we find $f(-2.5) \approx 6.25 + 5 + 3\cos(6.25 + 2.5) \approx 11.25 + 3\cos(8.75) \approx 11.25 + 3(0.684) \approx 11.25 + 2.052 \approx 13.302$ (rounded to three decimal places).We now have the point $(-2.5, 13.302)$ and the slope $-18.268$ . The equation of the tangent line in point-slope form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the point on the line.Substituting the values into the point-slope form, we get $y - 13.302 = -18.268(x - (-2.5))$ or $y - 13.302 = -18.268(x + 2.5)$ .To write the equation in slope-intercept form, we simplify the equation to $y = -18.268x - 18.268(2.5) + 13.302$ .
Convert to Slope-Intercept Form: Using a calculator, we find $f(-2.5) \approx 6.25 + 5 + 3\cos(6.25 + 2.5) \approx 11.25 + 3\cos(8.75) \approx 11.25 + 3(0.684) \approx 11.25 + 2.052 \approx 13.302$ (rounded to three decimal places).We now have the point $(-2.5, 13.302)$ and the slope $-18.268$ . The equation of the tangent line in point-slope form is $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is the point on the line.Substituting the values into the point-slope form, we get $y - 13.302 = -18.268(x - (-2.5))$ or $y - 13.302 = -18.268(x + 2.5)$ .To write the equation in slope-intercept form, we simplify the equation to $y = -18.268x - 18.268(2.5) + 13.302$ .Calculating the y-intercept, we get $y = -18.268x - 45.67 + 13.302$ , which simplifies to $(-2.5, 13.302)$ $0$ (rounded to three decimal places).