Calculus Test -...

If \(f(x)=\left\{\begin{array}{ll}\frac{\sin 3 x}{e^{2 x}-1}, & x \neq 0 \\ k-2, & x=0\end{array}\right.\) is continuous at \(x=0\), then \(k=\)?

Determine the value of \(k\) for which the following function is continuous at \(x=3\).

\(f(x)=\left\{\begin{array}{cc} \frac{x^2-9}{x-3}, & x \neq 3 \\ k, & x=3 \end{array}\right.\)

For the function \(f\) given by \(f(x)=x^2-6 x+8\), the value of \(f^{\prime}(5)-3 f^{\prime}(2)\) will be equals to:

If \(\sqrt{y}=\sin x+\cos x\), then \(\frac{d y}{d x}\) is:

Let \(f(x)=2 x^{3}+\frac{1}{x}\), then \(f'(1)\) is:

Determine the value of the constant \(k\) so that the function \(f(x)=\left\{\begin{array}{cl} k x^2, & \text { if } x \leq 2 \\ 3, & \text { if } x>2 \end{array}\right.\) is continuous.

Find the derivative of \(\frac{1}{3 x^{2}}\) with respect to \(x\).

If \(f(x)=\frac{\sin \left(e^{x-2}-1\right)}{\log (x-1)}, x \neq 2\) and \(f(x)=k\).Then, the value of k for which f will be continuous at x = 2 is:

The derivative of \(\cos ^{-1}\left(\frac{1}{2 x^{2}-1}\right)\) with respect to \(\sqrt{1-x^{2}}\) is:

Find \(\frac{\mathrm{dy}}{\mathrm{dx}}\), if \(\mathrm{y}=\tan ^{-1}\left[\frac{8 \mathrm{x}}{1-15 \mathrm{x}^{2}}\right]\).