The function f is defined by f(x)=x^(2)+1-cos(3x^(2)-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^(2)+1-cos(3x^(2)-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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Find Derivative: To find the equation of the tangent line at x = 0.5, we first need to find the derivative of the function f(x), which will give us the slope of the tangent line at any point x. The derivative of f(x) = x^2 + 1 - cos(3x^2 - 3) is f'(x) = 2x + sin(3x^2 - 3) * 6x, using the chain rule for derivatives.Evaluate 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) = 2(0.5) + sin(3(0.5)^2 - 3) * 6(0.5)Calculate f'(0.5): Calculating the value of f'(0.5) using a calculator and rounding to three decimal places: f'(0.5) = 1 + sin(3(0.25) - 3) * 3 f'(0.5) = 1 + sin(0.75 - 3) * 3 f'(0.5) = 1 + sin(-2.25) * 3 f'(0.5) ≈ 1 + sin(-2.25) * 3 ≈ 1 - 0.778 * 3 ≈ 1 - 2.334 ≈ -1.334Find y-coordinate: 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. f(0.5) = (0.5)^2 + 1 - cos(3(0.5)^2 - 3)Calculate f(0.5): Calculating the value of f(0.5) using a calculator and rounding to three decimal places: f(0.5) = 0.25 + 1 - cos(3(0.25) - 3) f(0.5) = 1.25 - cos(0.75 - 3) f(0.5) = 1.25 - cos(-2.25) f(0.5) ≈ 1.25 - cos(-2.25) ≈ 1.25 + 0.778 ≈ 2.028Use Point-Slope Form: Now we have the slope of the tangent line, m = -1.334, and a point on the tangent line, (0.5, 2.028). We can use the point-slope form of the equation of a line, y - y1 = m(x - x1), to write the equation of the tangent line.Plug in Values: Plugging in the values into the point-slope form: y - 2.028 = -1.334(x - 0.5)Simplify Equation: Simplifying the equation to get it into slope-intercept form, y = mx + b: y = -1.334x + (-1.334 * -0.5) + 2.028 y = -1.334x + 0.667 + 2.028 y = -1.334x + 2.695

The function $f$ is defined by $f(x)=x^{2}+1-\cos \left(3 x^{2}-3\right)$ . 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 f f is defined by f(x)=x2+1−cos⁡(3x2−3) f(x)=x^{2}+1-\cos \left(3 x^{2}-3\right) f(x)=x2+1−cos(3x2−3). 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 $f(x)=x^{2}+1-\cos \left(3 x^{2}-3\right)$ . 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: