The function f is defined by f(x)=x^(3)+2x+3sin(x^(2)+2). 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)+2x+3sin(x^(2)+2). 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(x): To find the equation of the tangent line at x = -0.5, we first need to calculate the derivative of f(x), which will give us the slope of the tangent line at any point x. The derivative of f(x) is f'(x) = 3x^2 + 2 + 3cos(x^2 + 2) * 2x (using the chain rule for the sine function).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. f'(-0.5) = 3(-0.5)^2 + 2 + 3cos((-0.5)^2 + 2) * 2(-0.5).Calculate cos(2.25): Using a calculator, we find that: f'(-0.5) = 3(0.25) + 2 + 3cos(0.25 + 2) * (-1) = 0.75 + 2 - 3cos(2.25) ≈ 2.75 - 3cos(2.25). Now we calculate the numerical value of cos(2.25) and then the entire expression.Find y-coordinate at x = -0.5: Assuming the calculator gives us the value of cos(2.25) rounded to three decimal places, we get: cos(2.25) ≈ -0.628 (rounded to three decimal places). Now we substitute this value into the expression for f'(-0.5): f'(-0.5) ≈ 2.75 - 3(-0.628) ≈ 2.75 + 1.884 ≈ 4.634. So, the slope of the tangent line at x = -0.5 is approximately 4.634.Use Point-Slope Form for Tangent Line: Next, we need to find the y-coordinate of the point on the graph of f(x) at x = -0.5 to use it in the point-slope form of the equation of the tangent line. We calculate f(-0.5) = (-0.5)^3 + 2(-0.5) + 3sin((-0.5)^2 + 2).Simplify Equation to Slope-Intercept Form: Using a calculator, we find that: f(-0.5) = -0.125 - 1 + 3sin(0.25 + 2) ≈ -1.125 + 3sin(2.25). Now we calculate the numerical value of sin(2.25) and then the entire expression.Calculate y-intercept: Assuming the calculator gives us the value of sin(2.25) rounded to three decimal places, we get: sin(2.25) ≈ 0.781 (rounded to three decimal places). Now we substitute this value into the expression for f(-0.5): f(-0.5) ≈ -1.125 + 3(0.781) ≈ -1.125 + 2.343 ≈ 1.218. So, the y-coordinate of the point on the graph of f(x) at x = -0.5 is approximately 1.218.Now we have the slope of the tangent line (m ≈ 4.634) and a point on the tangent line (-0.5, 1.218). We can use the point-slope form of the equation 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. Substituting the values we have, the equation becomes: y - 1.218 ≈ 4.634(x + 0.5).To write the equation in slope-intercept form (y = mx + b), we simplify the equation: y ≈ 4.634x + 4.634(0.5) + 1.218.Finally, we calculate the y-intercept (b): b ≈ 4.634(0.5) + 1.218 ≈ 2.317 + 1.218 ≈ 3.535. So, the equation of the tangent line is: y ≈ 4.634x + 3.535, rounded to three decimal places.

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