The function f is defined by f(x)=x^(3)+cos(2x^(2)+5). Use a calculator to write the equation of the line tangent to the graph of f when x=1. You should round all decimals to 3 places. Answer:

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The function  f is defined by  f(x)=x^(3)+cos(2x^(2)+5). Use a calculator to write the equation of the line tangent to the graph of  f when  x=1. You should round all decimals to 3 places. Answer:

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Calculate Derivative: To find the equation of the tangent line at x = 1, we need to calculate the derivative of f(x) to get f'(x), which will give us the slope of the tangent line at that point. f'(x) = d/dx [x^3 + cos(2x^2 + 5)] Using the power rule for the x^3 term and the chain rule for the cosine term, we get: f'(x) = 3x^2 - sin(2x^2 + 5) * d/dx [2x^2 + 5] f'(x) = 3x^2 - sin(2x^2 + 5) * (4x) Now we need to evaluate this derivative at x = 1.Evaluate Derivative at x=1: Plugging x = 1 into the derivative, we get: f'(1) = 3(1)^2 - sin(2(1)^2 + 5) * (4(1)) f'(1) = 3 - sin(7) * 4 Using a calculator to find sin(7) (in radians) and rounding to three decimal places, we get: sin(7) ≈ 0.657 (rounded to three decimal places) f'(1) = 3 - 0.657 * 4 f'(1) = 3 - 2.628 f'(1) = 0.372 (rounded to three decimal places) This is the slope of the tangent line at x = 1.Find y-coordinate at x=1: Next, we need to find the y-coordinate of the point on the graph of f(x) at x = 1 to use it in the point-slope form of the equation of a line. f(1) = 1^3 + cos(2(1)^2 + 5) f(1) = 1 + cos(9) Using a calculator to find cos(9) (in radians) and rounding to three decimal places, we get: cos(9) ≈ -0.911 (rounded to three decimal places) f(1) = 1 - 0.911 f(1) = 0.089 (rounded to three decimal places) Now we have the point (1, 0.089) on the graph of f(x).Use Point-Slope Form: We can now use the point-slope form of the equation of a line, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. Using the slope 0.372 and the point (1, 0.089), we get: y - 0.089 = 0.372(x - 1) To write the equation in slope-intercept form, we solve for y: y = 0.372x - 0.372 + 0.089 y = 0.372x - 0.283 (rounded to three decimal places) This is the equation of the tangent line at x = 1.

The function $f$ is defined by $f(x)=x^{3}+\cos \left(2 x^{2}+5\right)$ . Use a calculator to write the equation of the line tangent to the graph of $f$ when $x=1$ . 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+cos⁡(2x2+5) f(x)=x^{3}+\cos \left(2 x^{2}+5\right) f(x)=x3+cos(2x2+5). Use a calculator to write the equation of the line tangent to the graph of f f f when x=1 x=1 x=1. You should round all decimals to 333 places.\newlineAnswer:

The function $f$ is defined by $f(x)=x^{3}+\cos \left(2 x^{2}+5\right)$ . Use a calculator to write the equation of the line tangent to the graph of $f$ when $x=1$ . You should round all decimals to $3$ places. $\newline$ Answer: