The function $f$ is defined by $f(x)=x^{3}+2 \sin (3 x-3)$ . 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}+2 \sin (3 x-3)

. 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 and Slope: To find the equation of the tangent line to the graph of the function at $x = 0.5$ , we first need to calculate the derivative of the function, which will give us the slope of the tangent line at that point. $\newline$ The derivative of $f(x) = x^3 + 2\sin(3x - 3)$ is $f'(x) = 3x^2 + 6\cos(3x - 3)$ . $\newline$ Now we need to evaluate this derivative at $x = 0.5$ .
Evaluate Derivative at $x = 0.5$ : Calculating the derivative at $x = 0.5$ gives us $f'(0.5) = 3(0.5)^2 + 6\cos(3(0.5) - 3)$ . This simplifies to $f'(0.5) = 3(0.25) + 6\cos(1.5 - 3)$ . Further simplifying, we get $f'(0.5) = 0.75 + 6\cos(-1.5)$ . Using a calculator, we find $\cos(-1.5)$ and multiply it by $6$ , then add $0.75$ to get the slope of the tangent line.
Calculate Slope of Tangent Line: Using a calculator, $\cos(-1.5) \approx 0.0707$ (rounded to four decimal places for intermediate calculation). $\newline$ Now we calculate $6 \times 0.0707 + 0.75$ . $\newline$ This gives us $0.4242 + 0.75 = 1.1742$ . $\newline$ So, the slope of the tangent line at $x = 0.5$ is approximately $1.174$ (rounded to three decimal places).
Find y-coordinate at $x = 0.5$ : Next, we need to find the y-coordinate of the point on the graph of $f(x)$ at $x = 0.5$ to determine the point through which the tangent line passes. $\newline$ We calculate $f(0.5) = (0.5)^3 + 2\sin(3(0.5) - 3)$ . $\newline$ This simplifies to $f(0.5) = 0.125 + 2\sin(1.5 - 3)$ . $\newline$ Further simplifying, we get $f(0.5) = 0.125 + 2\sin(-1.5)$ . $\newline$ Using a calculator, we find $\sin(-1.5)$ and multiply it by $2$ , then add $0.125$ to get the y-coordinate.
Calculate y-coordinate: Using a calculator, $\sin(-1.5) \approx -0.9975$ (rounded to four decimal places for intermediate calculation). $\newline$ Now we calculate $2 \times (-0.9975) + 0.125$ . $\newline$ This gives us $-1.995 + 0.125 = -1.87$ . $\newline$ So, the y-coordinate of the point on the graph of $f(x)$ at $x = 0.5$ is approximately $-1.87$ (rounded to two decimal places).
Use Point-Slope Form: Now we have the slope of the tangent line $m = 1.174$ and a point on the tangent line $(0.5, -1.87)$ . We can use the point-slope form of the equation of a line to find the equation of the tangent line: $y - y_1 = m(x - x_1)$ . Plugging in our values, we get $y - (-1.87) = 1.174(x - 0.5)$ .
Simplify Equation: Simplifying the equation, we get $y + 1.87 = 1.174x - 0.587$ . Then, we subtract $1.87$ from both sides to get the final equation of the tangent line: $y = 1.174x - 0.587 - 1.87$ .
Final Tangent Line Equation: Combining like terms, we get $y = 1.174x - 2.457$ . This is the equation of the tangent line to the graph of $f(x)$ at $x = 0.5$ , rounded to three decimal places.