Étude 1-1: Defining a Function in the REPL

Create a project named formulas (see Creating a ClojureScript Project) and start a browser REPL (Read/Evaluate/Print/Loop). If you haven’t yet installed ClojureScript, follow the instructions in Setting Up Your ClojureScript Environment and create a project to work with. In the REPL, type the preceding distance function and test it.

Étude 1-2: Defining Functions in a Source File

Defining functions in the REPL is fine for a quick test, but it is not something you want to do on an application-level scale. Instead, you want to define the functions in a source file. In the formulas project, open the src/formulas/core.cljs file and create functions for these formulas:

• Distance equals one-half acceleration multplied by time squared: $d=\frac{1}{2}a{t}^2$
• Kinetic energy equals one-half the mass times velocity squared: $\mathrm{𝐾𝐸}=\frac{1}{2}m{v}^2$
• Centripetal acceleration is velocity squared over the radius: $a_c=\frac{v^2}{r}$

Here is some sample output. (in-ns 'formulas.core) switches you to that namespace so that you can type the function name without having to specify the module that it is in. If you update the source, (require 'formulas.core :reload) will recompile the code.

cljs.user=> (in-ns 'formulas.core)
nil
formulas.core=> (require 'formulas.core :reload)
nil
formulas.core=> (distance 9.8 5)
122.5
formulas.core=> (kinetic-energy 35 4)
280
formulas.core=> (centripetal 30 2)
450

See a suggested solution: Solution 1-2

Étude 1-3: Using def

The def special form lets you bind a symbol to a value. The symbol is globally available to all functions in the namespace where it is defined. Add a function named gravitational-force that calculates the gravitational force between two masses whose centers of mass are at a distance r from each other to your code:

$F=\frac{G{m}_1{m}_2}{r^2}$, where the gravitational constant $G=6.67384\times {10}^{-11}$. Use a def for the gravitational constant.

Here is the calculation for two masses of 100kg that are 5 meters apart:

formulas.core=> (gravitational-force 100 100 5)
2.67136e-8

(def x 5)
  (def x 6)
  (def x (+ 1 x))

See a suggested solution: Solution 1-3

Étude 1-4: Using let

To create local bindings of symbols to values within a function, you use let. The let is followed by a vector of symbol and value pairs.¹

In this étude, you will write a function named monthly-payment that calculates monthly payments on a loan. Your function will take the amount of the loan, the annual percentage rate, and the number of years of the loan as its three parameters. Calculate the monthly payment according to this formula:

$\mathrm{𝑝𝑎𝑦𝑚𝑒𝑛𝑡}=p\cdot \frac{r{(1+r)}^n}{{(1+r)}^n-1}$

• p is the principal (the amount of the loan)
• r is the monthly interest rate
• n is the number of months of the loan

Use let to make local bindings for:

• The monthly interest rate r, which equals the annual rate divided by 12
• The number of months n, which equals the number of years times 12
• The common sub-formula ${(1+r)}^n$

  To raise a number to a power, invoke the JavaScript pow function with code in this format:

  (.pow js/Math <em>number</em> <em>power</em>)
  ;; Thus, to calculate 3 to the fifth power:
  (.pow js/Math 3 5)
  

  You can also use this shorthand:

  (js/Math.pow 3 5)
  

  You will learn more about interacting with JavaScript in Interacting With JavaScript and Web Pages.

Here is some sample output for a loan of $1,000.00 at 5.5% for 15 years. You can also check the results of your function against the results of the PMT function in your favorite spreadsheet.

formulas.core=> (monthly-payment 1000 5.5 15). 
8.17083454621138

See a suggested solution: Solution 1-4

Étude 1-5: More Practice with def and let

Here’s a somewhat more complicated formula―determining the amount of sunlight in a day, given the day of year and the latitude of your location. Write a function named daylight with two parameters: a latitude in degrees and a Julian day. The function returns the number of minutes of sunshine for the day, using the formula explained at the “Ask Dr. Math” web site. The latitude is in degrees, but JavaScript’s trigonometric functions use radians, so you will need a function to convert degrees to radians, and I’ll give you that for free:

(defn radians
  "Convert degrees to radians"
  [degrees]
  (* (/ (.-PI js/Math) 180) degrees))

The expression (.-PI js/Math) gets the PI property of the JavaScript Math object.

• You will want a variable that holds the latitude converted to radians.
• Calculate $P=\arcsin (0.39795\cdot \cos (0.2163108+2\cdot \arctan (0.9671396\cdot \tan (0.00860\cdot (\mathrm{𝑑𝑎𝑦}-186)))))$
• Calculate $D=24-7.63944\cdot \arccos \left(\frac{\sin (0.01454)+\sin (\mathrm{𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒})\cdot \sin (p)}{\cos (\mathrm{𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒})\cdot \cos (p)}\right)$

The variable D holds the number of hours of daylight, so multiply that by 60 for your final result. If you feel that these formulas are a bit too complicated to type as single expressions (I certainly did!), break them down by using let for the parts.

On Mac OSX or Linux, you can get a Julian date with the date command:

$ date '+%j' # today
127
$ date -d '2015-09-15' '+%j' # arbitrary date
258

Your results should be very close to those generated by the National Oceanic and Atmospheric Administration spreadsheets, which use a far more complicated algorithm than the one given here.

See a suggested solution: Solution 1-5