Lecture 10 Exercise Solutions¶

Exercise 1¶

You will work with the following function for this exercise, \begin{align} f\left(x,y\right) = \exp\left(-\left(\sin\left(x\right) - \cos\left(y\right)\right)^{2}\right). \end{align}

1. Draw the computational graph for the function.
   • Note: This graph will have $2$ inputs.
2. Create the evaluation trace for this function.
3. Use the graph / trace to evaluate $f\left(\dfrac{\pi}{2}, \dfrac{\pi}{3}\right)$.
4. Compute $\dfrac{\partial f}{\partial x}\left(\dfrac{\pi}{2}, \dfrac{\pi}{3}\right)$ and $\dfrac{\partial f}{\partial y}\left(\dfrac{\pi}{2}, \dfrac{\pi}{3}\right)$ using the forward mode of AD.

Solution¶

Trace	Elementary Function	Current Value	Elementary Function Derivative	$\nabla_{x}$ Value	$\nabla_{y}$ Value
$x_{1}$	$x_{1}$	$\dfrac{\pi}{2}$	$\dot{x}_{1}$	$1$	$0$
$x_{2}$	$x_{2}$	$\dfrac{\pi}{3}$	$\dot{x}_{2}$	$0$	$1$
$v_{1}$	$\sin\left(x_{1}\right)$	$1$	$\cos\left(x_{1}\right)\dot{x}_{1}$	$0$	$0$
$v_{2}$	$\cos\left(x_{2}\right)$	$\dfrac{1}{2}$	$-\sin\left(x_{2}\right)\dot{x}_{2}$	$0$	$-\dfrac{\sqrt{3}}{2}$
$v_{3}$	$-v_{2}$	$-\dfrac{1}{2}$	$-\dot{v}_{2}$	$0$	$\dfrac{\sqrt{3}}{2}$
$v_{4}$	$v_{1} + v_{3}$	$\dfrac{1}{2}$	$\dot{v}_{1} + \dot{v}_{3}$	$0$	$\dfrac{\sqrt{3}}{2}$
$v_{5}$	$v_{4}^{2}$	$\dfrac{1}{4}$	$2v_{4}\dot{v}_{4}$	$0$	$\dfrac{\sqrt{3}}{2}$
$v_{6}$	$-v_{5}$	$-\dfrac{1}{4}$	$-\dot{v}_{5}$	$0$	$-\dfrac{\sqrt{3}}{2}$
$v_{7}$	$\exp\left(v_{6}\right)$	$\exp\left(-1/4\right)$	$\exp\left(v_{6}\right)\dot{v}_{6}$	$0$	$-\dfrac{\sqrt{3}}{2}\exp\left(-1/4\right)$

Exercise 2¶

\begin{align} f\left(x,y\right) = \begin{bmatrix} xy + \sin\left(x\right) \\ x + y + \sin\left(xy\right) \end{bmatrix}. \end{align}

Trace	Elem.	Val.	Elem. Der.	$\nabla_{x}$	$\nabla_{y}$
$x_{1}$	$x_{1}$	$a$	$\dot{x}_{1}$	$1$	$0$
$x_{2}$	$x_{2}$	$b$	$\dot{x}_{2}$	$0$	$1$
$v_{1}$	$x_{1}x_{2}$	$ab$	$x_{1}\dot{x}_{2} + x_{2}\dot{x}_{1}$	$b$	$a$
$v_{2}$	$x_{1} + x_{2}$	$a+b$	$\dot{x}_{1} + \dot{x}_{2}$	$1$	$1$
$v_{3}$	$\sin\left(x_{1}\right)$	$\sin\left(a\right)$	$\cos\left(x_{1}\right)\dot{x}_{1}$	$\cos\left(a\right)$	$0$
$v_{4}$	$\sin\left(v_{1}\right)$	$\sin\left(ab\right)$	$\cos\left(v_{1}\right)\dot{v}_{1}$	$b\cos\left(ab\right)$	$a\cos\left(ab\right)$
$v_{5}$	$v_{1} + v_{3}$	$ab + \sin\left(a\right)$	$\dot{v_{1}} + \dot{v_{3}}$	$b + \cos\left(a\right)$	$a$
$v_{6}$	$v_{2} + v_{4}$	$a+b + \sin\left(ab\right)$	$\dot{v}_{2} + \dot{v}_{4}$	$1 + b\cos\left(ab\right)$	$1 + a\cos\left(ab\right)$