The function f is defined by f(x)=x^(2)+x-2sin(2x). Use a calculator to write the equation of the line tangent to the graph of f when x=3. You should round all decimals to 3 places. Answer:

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The function  f is defined by  f(x)=x^(2)+x-2sin(2x). Use a calculator to write the equation of the line tangent to the graph of  f when  x=3. 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 at x = 3, we first need to calculate the derivative of the function f(x) to find the slope of the tangent line at that point. The derivative of f(x) = x^2 + x - 2sin(2x) is f'(x) = 2x + 1 - 4cos(2x). Now we will calculate f'(3) to get the slope of the tangent line at x = 3.Calculate f'(3): Using a calculator, we find the value of f'(3) = 2(3) + 1 - 4cos(2*3). This gives us f'(3) = 6 + 1 - 4cos(6). Now we will calculate the numerical value of cos(6) and then find the value of f'(3).Find y-coordinate at x=3: Assuming the calculator is set to radian mode, we find that cos(6) ≈ 0.960 (rounded to three decimal places). Now we calculate f'(3) = 6 + 1 - 4(0.960). This gives us f'(3) = 7 - 3.840 = 3.160 (rounded to three decimal places). The slope of the tangent line at x = 3 is 3.160.Use Point-Slope Form: Next, we need to find the y-coordinate of the point on the graph of f(x) at x = 3 to use it in the point-slope form of the equation of the tangent line. We calculate f(3) = 3^2 + 3 - 2sin(2*3).Write Equation in Slope-Intercept Form: Using the calculator, we find that sin(6) ≈ -0.279 (rounded to three decimal places). Now we calculate f(3) = 9 + 3 - 2(-0.279). This gives us f(3) = 12 + 0.558 = 12.558 (rounded to three decimal places). The y-coordinate of the point on the graph of f(x) at x = 3 is 12.558.We now have the slope of the tangent line (m = 3.160) and a point on the tangent line (3, 12.558). Using the point-slope form of the equation of a line, y - y1 = m(x - x1), we can write the equation of the tangent line. Substituting the values, we get y - 12.558 = 3.160(x - 3).To write the equation in slope-intercept form, we simplify the equation: y = 3.160(x - 3) + 12.558. Expanding this, we get y = 3.160x - 9.480 + 12.558. Finally, we combine like terms to get y = 3.160x + 3.078. This is the equation of the tangent line to the graph of f(x) at x = 3, rounded to three decimal places.

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

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