The function f is defined by f(x)=x^(3)+2sin(3x-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. Answer:

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The function  f is defined by  f(x)=x^(3)+2sin(3x-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. Answer:

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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. The derivative of f(x) = x^3 + 2sin(3x - 3) is f'(x) = 3x^2 + 6cos(3x - 3). 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 + 6cos(3(0.5) - 3). This simplifies to f'(0.5) = 3(0.25) + 6cos(1.5 - 3). Further simplifying, we get f'(0.5) = 0.75 + 6cos(-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) ≈ 0.0707 (rounded to four decimal places for intermediate calculation). Now we calculate 6 * 0.0707 + 0.75. This gives us 0.4242 + 0.75 = 1.1742. 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. We calculate f(0.5) = (0.5)^3 + 2sin(3(0.5) - 3). This simplifies to f(0.5) = 0.125 + 2sin(1.5 - 3). Further simplifying, we get f(0.5) = 0.125 + 2sin(-1.5). 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) ≈ -0.9975 (rounded to four decimal places for intermediate calculation). Now we calculate 2 * (-0.9975) + 0.125. This gives us -1.995 + 0.125 = -1.87. 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 - y1 = m(x - x1). 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.

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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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: