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A function \( f(x) \) is defined as\[ f(x) = \begin{cases} 2x + 3 & \text{for } x < 1 \\ 4 & \text{for } x = 1 \\ 6x - 1 & \text{for } x > 1 \end{cases} \] Is the function \( f(x) \) continuous at \( x = 1 \)? If not, how can you make it continuous at \( x = 1 \)?
Find the derivative of \( e^{-2x} \) using the first principle. \[ -\sum e^{-\frac{x^2}{2}}. \]
Evaluate \[ \lim_{x \to \frac{\pi}{4}} \frac{\sin x - \cos x}{x - \frac{\pi}{4}} \]
Find area of the region enclosed by the parabola \( y = 2x^2 \) and line \( y = x \). Evaluate a) \[ \int_{0}^{2} \frac{dx}{\sqrt{x^2 + 4^2}} \] b) \[ \int x^2 e^x \, dx \]
State Mean Value Theorem. Give its geometrical meaning. Verify the Mean Value Theorem for the function \( f(x) = x^3 + x^2 - 6x \) in the interval \([-1, 4]\).
Examine the consistency of the system. Solve it by using Gauss elimination method. \[ \begin{cases} 3x + y + z = 5 \\ 3x - 4y + z = -2 \\ 3x + y - 3z = -1 \end{cases} \]
Using simplex method, find the optimal solution of the following linear programming problem. Minimize \( Z = 10x + 15y \). Subject to \[ \begin{cases} x + y \geq 8 \\ 5x + 3y \geq 30 \\ x \geq 0, \, y \geq 0 \end{cases} \]
Attempt any TWO questions
[2x10=20]Solve the following: a) \( \frac{dy}{dx} - 3y = x^2 \)
b) \( x\,dy - y\,dx = x^3 y\,dy \)
a) Use Simpson's \( \frac{1}{3} \) Rule to evaluate \[ \int_0^1 \frac{1}{1+x^2} dx \] taking \( n = 4 \). Also find the error. b) A man who has 130 m of fencing material wishes to enclose a rectangular garden. Find the maximum area he can enclose.
Compute the approximate value of the integral $$\int \frac{1}{1 + x^2}\,dx$$ by using the composite trapezoidal rule with three points, and compare the result with the actual value. Determine the error formula and numerically verify an upper bound on it.