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

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The function  f is defined by  f(x)=x^(2)-5cos(2x). Use a calculator to write the equation of the line tangent to the graph of  f when  x=3. 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 = 3, 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) = 2x - 5 * derivative of cos(2x). The derivative of cos(2x) with respect to x is -sin(2x) * 2 (using the chain rule). Therefore, f'(x) = 2x - 5 * (-2sin(2x)) = 2x + 10sin(2x).Evaluate Derivative at x=3: Now we need to evaluate the derivative at x = 3 to find the slope of the tangent line. So we calculate f'(3) = 2(3) + 10sin(2*3). Using a calculator, we find that sin(6) is approximately -0.279. Therefore, f'(3) ≈ 6 + 10(-0.279) = 6 - 2.79 = 3.21.Find y-coordinate at x=3: Next, we need to find the y-coordinate of the point on the graph of f where x = 3. We do this by evaluating f(3) = 3^2 - 5cos(2*3). Using a calculator, we find that cos(6) is approximately 0.960. Therefore, f(3) ≈ 9 - 5(0.960) = 9 - 4.8 = 4.2.Write Equation of Tangent Line: With the slope of the tangent line (m = 3.21) and the point on the graph (3, 4.2), we can use the point-slope form of the equation of a line to write 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. Plugging in our values, we get y - 4.2 = 3.21(x - 3).Convert to Slope-Intercept Form: Finally, we simplify the equation to put it in slope-intercept form (y = mx + b). We distribute the slope on the right side of the equation and then add 4.2 to both sides to solve for y. y - 4.2 = 3.21x - 9.63. Adding 4.2 to both sides gives us y = 3.21x - 9.63 + 4.2. Simplifying further, y = 3.21x - 5.43.

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

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