A function \(f(x)\) is defined as
\(f(x)=2x+3\) for \(x<1\)
\(f(x)=4\) for \(x=1\)
\(f(x)=6x-1\) for \(x>1\)
Is the function \(f(x)\) is continuous at \(x=1\)?
If not, how can you make it continuous at \(x=1\)?
Evaluate the limit, \(\lim_{x \to \infty}\left(\frac{x\cos\theta-\theta\cos x}{x-\theta}\right)\)
Find the maximum and minimum values of the function \(f(x)=4x^3-15x^2+12x-1\). Also, find the point of inflection.
Find \(\frac{dy}{dx}\) if
(a) \(x=t^2-1,\ y=t^4-1\)
(b) \(x^2+y^2=\sin xy\)
Evaluate \(\int_0^2 \sqrt{1+x^3}\,dx\) by using Simpson's \(\frac{1}{3}\) rule by taking \(n=4\).
Find the derivative of the function \(f(x)=\frac{1}{\sqrt{x}}\) by using first principle.
Solve the differential equations
\(2\frac{dy}{dx}=2\frac{y}{x}+\frac{y^2}{x^2}\)
Attempt any TWO questions
[2x10=20]State Rolle's theorem and Lagrange's mean value theorem with their geometrical interpretation. Verify Rolle's theorem for the function \(f(x)=\sin x,\ x\in[0,\pi]\). Also find a point on the curve represented by given function where the target is parallel to x-axis.
Using the simplex method, find the optimal solution of the following linear programming problem. Maximize, \(z=15x+12y\)
Subject to, \(2x+3y\leq21\)
\(3x+2y\leq24\)
\(x,y\geq0\)
(a) State L' Hospital Rule. Use it to evaluate \(\lim_{x \to 0}\left(\frac{xe^x-\ln(1+x)}{x^2}\right)\)
(b) Solve the following system of equations by Gauss-Seidel method.
\(x-2y+3z=1\)
\(3x-y+4z=3\)
\(2x+y-2z=-1\)