The function f is defined by f(x)=x^(3)+4-4sin(x^(2)). Use a calculator to write the equation of the line tangent to the graph of f when x=2.5. You should round all decimals to 3 places. Answer:

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The function  f is defined by  f(x)=x^(3)+4-4sin(x^(2)). Use a calculator to write the equation of the line tangent to the graph of  f when  x=2.5. 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 = 2.5, 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) = 3x^2 - 8xsin(x^2)cos(x^2) (using the chain rule and product rule for differentiation).Evaluate Derivative at x = 2.5: Now we need to evaluate the derivative at x = 2.5 to find the slope of the tangent line at that point. So we calculate f'(2.5) = 3(2.5)^2 - 8(2.5)sin((2.5)^2)cos((2.5)^2). Using a calculator, we find the value of f'(2.5) and round it to three decimal places.Calculate Slope of Tangent Line: After calculating, we find that f'(2.5) ≈ 3(2.5)^2 - 8(2.5)sin((2.5)^2)cos((2.5)^2) ≈ 18.750 - 8(2.5)sin(6.25)cos(6.25) ≈ 18.750 - 20sin(6.25)cos(6.25). We need to use a calculator to find the exact value of sin(6.25) and cos(6.25) and then multiply by -20 and add to 18.750.Find y-coordinate at x = 2.5: Using a calculator, we find that sin(6.25) ≈ -0.003 and cos(6.25) ≈ 1.000. Therefore, -20sin(6.25)cos(6.25) ≈ -20(-0.003)(1.000) ≈ 0.060. Adding this to 18.750 gives us the slope of the tangent line: 18.750 + 0.060 ≈ 18.810.Write Equation of Tangent Line: Next, we need to find the y-coordinate of the point on the graph of f where x = 2.5. We do this by evaluating f(2.5) = (2.5)^3 + 4 - 4sin((2.5)^2). Using a calculator, we find f(2.5) and round it to three decimal places.Simplify Equation to Slope-Intercept Form: After calculating, we find that f(2.5) ≈ (2.5)^3 + 4 - 4sin(6.25) ≈ 15.625 + 4 - 4(-0.003) ≈ 19.625 + 0.012 ≈ 19.637.Now we have the slope of the tangent line, m = 18.810, and a point on the tangent line, (2.5, 19.637). We can use the point-slope form of a line, y - y1 = m(x - x1), to write the equation of the tangent line. Substituting the values, we get y - 19.637 = 18.810(x - 2.5).Finally, we simplify the equation to get it into slope-intercept form, y = mx + b. We distribute the slope and add 19.637 to both sides to find the y-intercept, b. y = 18.810x - 18.810(2.5) + 19.637.After simplifying, we find the equation of the tangent line to be y ≈ 18.810x - 47.018 + 19.637, which simplifies to y ≈ 18.810x - 27.381 (rounded to three decimal places).

The function $f$ is defined by $f(x)=x^{3}+4-4 \sin \left(x^{2}\right)$ . Use a calculator to write the equation of the line tangent to the graph of $f$ when $x=2.5$ . You should round all decimals to $3$ places. $\newline$ Answer:

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The function $f$ is defined by $f(x)=x^{3}+4-4 \sin \left(x^{2}\right)$ . Use a calculator to write the equation of the line tangent to the graph of $f$ when $x=2.5$ . You should round all decimals to $3$ places. $\newline$ Answer: