Calculus

To find the limit, we can factor the numerator and cancel out the common factor (x - 2): (\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} \frac{(x + 2)(x - 2)}{x - 2} = \lim_{x \to 2} (x + 2) = 2 + 2 = 4).

The derivative of (f(x)) can be found using the power rule of differentiation: (f'(x) = \frac{d}{dx}(x^3 - 2x^2 + 3x - 5) = 3x^2 - 4x + 3).

To evaluate the integral, we can use the power rule of integration: (\int_{0}^{1} x^2 dx = \left[\frac{x^3}{3}\right]_{0}^{1} = \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3}).

The antiderivative of (f(x) = \sin(x)) can be found using the integration rule for sine: (\int \sin(x) dx = -\cos(x) + C), where (C) is the constant of integration.

To find the area under the curve, we can use the definite integral: (\int_{0}^{2} x^2 dx = \left[\frac{x^3}{3}\right]_{0}^{2} = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3}).

To find the equation of the tangent line, we need to find the derivative of the function and evaluate it at the given point. The derivative is (f'(x) = 3x^2 - 4x + 3), and at (x = 1), the slope is (f'(1) = 3(1)^2 - 4(1) + 3 = 2). Using the point-slope form, the equation of the tangent line is (y - (-3) = 2(x - 1)), which simplifies to (y = 4x - 7).

To find the volume of the solid, we can use the method of cylindrical shells. The radius of each shell is (r = x), and the height is (h = 4 - x^2 - x^2 = 4 - 2x^2). The volume of each shell is (dV = 2\pi r h dx = 2\pi x (4 - 2x^2) dx). Integrating this expression from (x = 0) to (x = 2), we get the total volume: (V = \int_{0}^{2} 2\pi x (4 - 2x^2) dx = \frac{64}{3}\pi).

The normal line to a curve at a given point is perpendicular to the tangent line at that point. Since the slope of the tangent line is 2 at the point ((1, -3)), the slope of the normal line is (-\frac{1}{2}). Using the point-slope form, the equation of the normal line is (y - (-3) = -\frac{1}{2}(x - 1)), which simplifies to (y = -\frac{1}{2}x + \frac{1}{2}).

To find the indefinite integral, we can use partial fraction decomposition to rewrite the integrand as (\frac{x^2 + 2x - 3}{x - 1} = x + 3 + \frac{1}{x - 1}). Then, we can integrate each term separately: (\int (x + 3 + \frac{1}{x - 1}) dx = \frac{x^2}{2} + 3x + \ln|x - 1| + C), where (C) is the constant of integration.

To find the equation of the curve, we can integrate the given differential equation: (\int \frac{dy}{dx} dx = \int \frac{x^2 + 1}{y} dx). This gives us (y = \int \frac{x^2 + 1}{y} dx = \int \frac{x^2}{y} dx + \int \frac{1}{y} dx = \frac{x^3}{3y} + \ln|y| + C). Simplifying this expression, we get (y^2 = x^3 + x + C), where (C) is the constant of integration.

To find the area of the region, we can integrate the difference between the two functions with respect to (x): (\int_{0}^{2} (x^2 - 2x - x) dx = \int_{0}^{2} (x^2 - 3x) dx = \left[\frac{x^3}{3} - \frac{3x^2}{2}\right]_{0}^{2} = \frac{4}{3}).

To find the equation of the tangent plane, we need to find the partial derivatives of the function (f(x, y) = x^2 + y^2) and evaluate them at the given point. The partial derivatives are (f_x(x, y) = 2x) and (f_y(x, y) = 2y). At the point ((1, 2, 5)), the partial derivatives are (f_x(1, 2) = 2) and (f_y(1, 2) = 4). Using the point-normal form, the equation of the tangent plane is (z - 5 = 2(x - 1) + 4(y - 2)), which simplifies to (z = 5 + 2x + 4y).

To find the volume of the solid, we can use the method of cylindrical shells. The radius of each shell is (r = x), and the height is (h = 4 - x^2 - x^2 = 4 - 2x^2). The volume of each shell is (dV = 2\pi r h dx = 2\pi x (4 - 2x^2) dx). Integrating this expression from (x = 0) to (x = 2), we get the total volume: (V = \int_{0}^{2} 2\pi x (4 - 2x^2) dx = \frac{64}{3}\pi).

To find the equation of the curve, we can use the formula for curvature: (\kappa = \frac{|y''|}{\sqrt{1 + (y')^2}}). Substituting the given expression for (\kappa), we get (\frac{2}{\sqrt{x^2 + y^2}} = \frac{|y''|}{\sqrt{1 + (y')^2}}). Squaring both sides and simplifying, we get (4(1 + (y')^2) = y''^2). This is a differential equation that can be solved to obtain the equation of the curve. The solution is (y = \sin(x) + C), where (C) is the constant of integration.