The function f is defined by f(x)=x^(3)+3-5sin(x^(2)). 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)+3-5sin(x^(2)). 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^3 + 3 - 5sin(x^2) using the power rule and the chain rule is: f'(x) = 3x^2 - 10x * cos(x^2). 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) = 3(1)^2 - 10(1) * cos(1^2) = 3 - 10cos(1). Using a calculator, we find that cos(1) is approximately 0.540 (rounded to three decimal places). So, f'(1) ≈ 3 - 10 * 0.540 = 3 - 5.4 = -2.4. The slope of the tangent line at x = 1 is approximately -2.4.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. This is done by evaluating the original function at x = 1. f(1) = 1^3 + 3 - 5sin(1^2) = 1 + 3 - 5sin(1). Using a calculator, we find that sin(1) is approximately 0.841 (rounded to three decimal places). So, f(1) ≈ 1 + 3 - 5 * 0.841 = 4 - 4.205 = -0.205. The y-coordinate of the point on the graph at x = 1 is approximately -0.205.Use Point-Slope Form: Now we have the slope of the tangent line (-2.4) and a point on the line (1, -0.205). We can use the point-slope form of the equation of a line to find the equation of the tangent line: 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 - (-0.205) = -2.4(x - 1).Simplify and Finalize Equation: Simplifying the equation, we get: y + 0.205 = -2.4x + 2.4. Subtracting 0.205 from both sides to get y by itself, we have: y = -2.4x + 2.4 - 0.205. Combining like terms, we get: y = -2.4x + 2.195. This is the equation of the tangent line, rounded to three decimal places.

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