The function $f$ is defined by $f(x)=x^{3}+4 \cos \left(2 x^{2}\right)$ . Use a calculator to write the equation of the line tangent to the graph of $f$ when $x=0.5$ . 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^{3}+4 \cos \left(2 x^{2}\right)

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

f

when

x=0.5

. You should round all decimals to

3

places.

\newline

Answer:

Calculate Derivative of f: To find the equation of the tangent line to the graph of $f$ at $x = 0.5$ , we need to calculate the derivative of $f$ , which will give us the slope of the tangent line at that point. The derivative of $f$ with respect to $x$ is $f'(x) = 3x^2 - 8x\sin(2x^2)$ .
Evaluate Derivative at $x = 0.5$ : Now we need to evaluate the derivative at $x = 0.5$ to find the slope of the tangent line at that point. Plugging $x = 0.5$ into the derivative, we get $f'(0.5) = 3(0.5)^2 - 8(0.5)\sin(2(0.5)^2)$ .
Find y-coordinate at x = $0.5$ : Using a calculator, we calculate $f'(0.5) = 3(0.25) - 8(0.5)\sin(2(0.25)) = 0.75 - 4\sin(0.5)$ . We then round the result to three decimal places.
Calculate Slope of Tangent Line: After calculating the above expression using a calculator, we find that $f'(0.5) \approx 0.75 - 4\sin(0.5) \approx 0.75 - 4(0.479) \approx 0.75 - 1.916 \approx -1.166$ (rounded to three decimal places).
Use Point-Slope Form: Next, we need to find the $y$ -coordinate of the point on the graph of $f$ where $x = 0.5$ . We do this by evaluating $f(0.5) = (0.5)^3 + 4\cos(2(0.5)^2)$ .
Write Equation in Slope-Intercept Form: Using a calculator, we calculate $f(0.5) = (0.125) + 4\cos(2(0.25)) = 0.125 + 4\cos(0.5)$ . We then round the result to three decimal places.
Combine Constants: After calculating the above expression using a calculator, we find that $f(0.5) \approx 0.125 + 4\cos(0.5) \approx 0.125 + 4(0.877) \approx 0.125 + 3.508 \approx 3.633$ (rounded to three decimal places).
Combine Constants: After calculating the above expression using a calculator, we find that $f(0.5) \approx 0.125 + 4\cos(0.5) \approx 0.125 + 4(0.877) \approx 0.125 + 3.508 \approx 3.633$ (rounded to three decimal places).Now we have the slope of the tangent line, $m = -1.166$ , and a point on the tangent line, $(0.5, 3.633)$ . We can use the point-slope form of a line to write the equation of the tangent line: $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is the point on the line.
Combine Constants: After calculating the above expression using a calculator, we find that $f(0.5) \approx 0.125 + 4\cos(0.5) \approx 0.125 + 4(0.877) \approx 0.125 + 3.508 \approx 3.633$ (rounded to three decimal places).Now we have the slope of the tangent line, $m = -1.166$ , and a point on the tangent line, $(0.5, 3.633)$ . We can use the point-slope form of a line to write the equation of the tangent line: $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is the point on the line.Plugging in the values, we get the equation of the tangent line as $y - 3.633 = -1.166(x - 0.5)$ .
Combine Constants: After calculating the above expression using a calculator, we find that $f(0.5) \approx 0.125 + 4\cos(0.5) \approx 0.125 + 4(0.877) \approx 0.125 + 3.508 \approx 3.633$ (rounded to three decimal places).Now we have the slope of the tangent line, $m = -1.166$ , and a point on the tangent line, $(0.5, 3.633)$ . We can use the point-slope form of a line to write the equation of the tangent line: $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is the point on the line.Plugging in the values, we get the equation of the tangent line as $y - 3.633 = -1.166(x - 0.5)$ .To write the equation in slope-intercept form, we simplify the equation: $y = -1.166x + 0.583 + 3.633$ .
Combine Constants: After calculating the above expression using a calculator, we find that $f(0.5) \approx 0.125 + 4\cos(0.5) \approx 0.125 + 4(0.877) \approx 0.125 + 3.508 \approx 3.633$ (rounded to three decimal places).Now we have the slope of the tangent line, $m = -1.166$ , and a point on the tangent line, $(0.5, 3.633)$ . We can use the point-slope form of a line to write the equation of the tangent line: $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is the point on the line.Plugging in the values, we get the equation of the tangent line as $y - 3.633 = -1.166(x - 0.5)$ .To write the equation in slope-intercept form, we simplify the equation: $y = -1.166x + 0.583 + 3.633$ .Finally, we combine the constants to get the equation of the tangent line: $y = -1.166x + 4.216$ (rounded to three decimal places).