The function f is defined by f(x)=x^(3)-5+cos(2x). 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)-5+cos(2x). 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 and Slope: To find the equation of the tangent line at x = 1, we need to calculate the derivative of f(x) to find the slope of the tangent line at that point. The derivative of f(x) is f'(x) = 3x^2 - sin(2x) * 2, using the power rule for x^3, the constant rule for -5, and the chain rule for cos(2x). Now we need to evaluate f'(x) at x = 1 to find the slope of the tangent line.Evaluate f'(1) for Slope: Calculating f'(1) gives us the slope of the tangent line at x = 1. f'(1) = 3(1)^2 - sin(2*1) * 2 = 3 - sin(2) * 2. Using a calculator, we find that sin(2) is approximately 0.909, so f'(1) ≈ 3 - 0.909 * 2.Find y-coordinate at x=1: Now we perform the calculation for f'(1). f'(1) ≈ 3 - 0.909 * 2 = 3 - 1.818 = 1.182 (rounded to three decimal places). So, the slope of the tangent line at x = 1 is approximately 1.182.Use Point-Slope Form: Next, we need to find the y-coordinate of the point on the graph of f(x) where x = 1. This is done by evaluating f(1). f(1) = 1^3 - 5 + cos(2*1) = 1 - 5 + cos(2). Using a calculator, we find that cos(2) is approximately 0.540, so f(1) ≈ 1 - 5 + 0.540.Simplify Tangent Line Equation: Now we perform the calculation for f(1). f(1) ≈ 1 - 5 + 0.540 = -4 + 0.540 = -3.460 (rounded to three decimal places). So, the y-coordinate of the point on the graph of f(x) at x = 1 is approximately -3.460.Find y-intercept: With the slope of the tangent line and the point (1, -3.460), we can use the point-slope form of the equation of a line to find the equation of the tangent line. The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point on the line. Substituting the values we have, the equation becomes y - (-3.460) = 1.182(x - 1).Simplifying the equation of the tangent line, we get: y + 3.460 = 1.182x - 1.182. Now we subtract 3.460 from both sides to solve for y. y = 1.182x - 1.182 - 3.460.Finally, we perform the subtraction to find the y-intercept of the tangent line. y = 1.182x - 1.182 - 3.460 = 1.182x - 4.642 (rounded to three decimal places). This is the equation of the tangent line at x = 1 for the function f(x).

The function $f$ is defined by $f(x)=x^{3}-5+\cos (2 x)$ . 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 $f$ is defined by $f(x)=x^{3}-5+\cos (2 x)$ . 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: