The function f is defined by f(x)=x^(2)+5+3sin(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^(2)+5+3sin(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: To find the equation of the tangent line at x = 1, we first need to calculate the derivative of the function f(x) to find the slope of the tangent line at that point. The derivative of f(x) = x^2 + 5 + 3sin(2x) is f'(x) = 2x + 3 * 2cos(2x) = 2x + 6cos(2x). Now we need to evaluate this derivative at x = 1.Evaluate Derivative at x=1: Evaluating the derivative at x = 1 gives us f'(1) = 2(1) + 6cos(2(1)) = 2 + 6cos(2). Using a calculator, we find that cos(2) is approximately 0.540. So, f'(1) ≈ 2 + 6 * 0.540 = 2 + 3.24 = 5.24. We round this to three decimal places, so the slope of the tangent line at x = 1 is approximately 5.240.Find y-coordinate at x=1: Next, we need to find the y-coordinate of the point on the graph of f(x) where x = 1. We do this by evaluating the original function at x = 1: f(1) = 1^2 + 5 + 3sin(2*1) = 1 + 5 + 3sin(2). Using a calculator, we find that sin(2) is approximately 0.909. So, f(1) ≈ 1 + 5 + 3 * 0.909 = 6 + 2.727 = 8.727. We round this to three decimal places, so the y-coordinate is approximately 8.727.Write Point-Slope Equation: Now we have the slope of the tangent line (m = 5.240) and a point on the tangent line (x = 1, y ≈ 8.727). We can use the point-slope form of a line to write the equation of the tangent line: y - y1 = m(x - x1). Substituting the known values, we get y - 8.727 = 5.240(x - 1).Convert to Slope-Intercept Form: To write the equation in slope-intercept form (y = mx + b), we simplify the equation from the previous step: y = 5.240(x - 1) + 8.727 y = 5.240x - 5.240 + 8.727 y = 5.240x + 3.487 We round the y-intercept to three decimal places, so the equation of the tangent line is y ≈ 5.240x + 3.487.

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