The function f is defined by f(x)=x^(3)+1+2sin(2x). 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)+1+2sin(2x). 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 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 + 4cos(2x).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. So we calculate f'(0.5) = 3(0.5)^2 + 4cos(2*0.5).Find y-coordinate at x = 0.5: Using a calculator, we find that f'(0.5) = 3(0.25) + 4cos(1) which is approximately 0.75 + 4(0.540) after rounding to three decimal places. This gives us f'(0.5) ≈ 0.75 + 2.160 = 2.910.Determine Slope and Point: Next, we need to find the y-coordinate of the point on the graph of f at x = 0.5. We do this by evaluating f(0.5) = (0.5)^3 + 1 + 2sin(2*0.5).Write Equation in Point-Slope Form: Using a calculator, we find that f(0.5) = 0.125 + 1 + 2sin(1) which is approximately 0.125 + 1 + 2(0.841) after rounding to three decimal places. This gives us f(0.5) ≈ 0.125 + 1 + 1.682 = 2.807.Convert to Slope-Intercept Form: Now we have the slope of the tangent line, m = 2.910, and a point on the tangent line, (0.5, 2.807). We can use the point-slope form of a line to write the equation of the tangent line: y - y1 = m(x - x1), where (x1, y1) is the point on the line.Final Equation: Substituting the values we have, the equation of the tangent line is y - 2.807 = 2.910(x - 0.5).To write the equation in slope-intercept form, we distribute the slope on the right side and add 2.807 to both sides: y = 2.910x - 1.455 + 2.807.Simplifying the equation, we get y = 2.910x + 1.352 after rounding to three decimal places.

The function $f$ is defined by $f(x)=x^{3}+1+2 \sin (2 x)$ . 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 f f is defined by f(x)=x3+1+2sin⁡(2x) f(x)=x^{3}+1+2 \sin (2 x) f(x)=x3+1+2sin(2x). Use a calculator to write the equation of the line tangent to the graph of f f f when x=0.5 x=0.5 x=0.5. You should round all decimals to 333 places.\newlineAnswer:

The function $f$ is defined by $f(x)=x^{3}+1+2 \sin (2 x)$ . 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: