Question

Question 1. Compute the derivative of the following functions.

(a) \(f(x)=x^{3}-\frac{2}{\sqrt{x}}+4\)

(b) \(f(x)=2^{3 x-1}\)

(c) \(f(x)=\ln \left(5 x^{2}+1\right)\)

(d) \(f(x)=\frac{\tan (x)}{x^{2}+1}\)

(e) \(f(x)=e^{x^{2}} \cdot \arctan (2 x)\)

(f) \(f(x)=\sin (x)^{2} \cdot\left(\tan (x)+\cos (x)^{2}\right)\).

Question 2. In geometry, the folium of Descartes is a curve given by the equation

$$ x^{3}+y^{3}-3 a x y=0 $$

Here, \(a\) is a constant.

The curve was first proposed by Descartes in 1638 . Its claim to fame lies in an incident in the development of calculus. Descartes challenged Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do.

It is now the time for you to take on Descartes' challenge.

(a) Find the derivative \(\frac{d y}{d x}\) at any point \(\left(x_{0}, y_{0}\right)\) on this curve. Here, you may assume \(\left(x_{0}, y_{0}\right) \neq(0,0)\).

(b) Assuming \(a=\frac{3}{2}\). Verify that \((2,1)\) is a point on the curve.

(c) Assuming \(a=\frac{3}{2}\). Find the equation of tangent line at \((2,1)\).

Question 3. Let \(f(x)=x^{3}+2 x-1\), and we let \(g(x)\) be the inverse function for \(f(x)^{2}\)

(a) Find the equation of the tangent line for \(f(x)\) at \(x=1\).

(b) Find the value of \(g(2)\).

(c) Find the equation of the tangent line for \(g(x)\) at \(x=2\).

(d) (Not to be handed in) Plot the two tangent lines you found for \(f(x), g(x)\). Do they look like what you expect them to be (i.e. are they reflection over \(y=x ?\) )

Question 4. The standard normal distribution (more commonly known as the bell curve) is one of the most important distributions in the probability and statistics. Its cumulative distribution function is often denoted by \(\Phi(x)\), and there is no way to express \(\Phi(x)\) by elementary functions. However, we know that its derivative:

$$ \Phi^{\prime}(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{x^{2}}} $$

We also know that \(\Phi(0)=\frac{1}{2}\).

(a) Let \(f(x)=\Phi\left(x^{2}-1\right)\). Find \(f^{\prime}(1)\).

(b) Let \(g(x)=x \cdot \Phi\left(x^{2}-1\right)\). Find \(g^{\prime}(1)\).

(c) Let \(h(x)=\left(\Phi\left(x^{2}-1\right)\right)^{2}\). Find \(h^{\prime}(1)\).

Answer 1