From 0abd77a749c9571a1f63406052141cf8795c70d5 Mon Sep 17 00:00:00 2001
From: genedan <genedan@gmail.com>
Date: Thu, 30 Jul 2020 16:48:28 -0500
Subject: [PATCH] write accumulation function tutorial

---
 docs/usage/growth/accumulation.rst | 68 +++++++++++++++++++++++++++++-
 1 file changed, 67 insertions(+), 1 deletion(-)

diff --git a/docs/usage/growth/accumulation.rst b/docs/usage/growth/accumulation.rst
index ac5b05b..1851086 100644
--- a/docs/usage/growth/accumulation.rst
+++ b/docs/usage/growth/accumulation.rst
@@ -1,3 +1,69 @@
 ========================
 Accumulation Functions
-========================
\ No newline at end of file
+========================
+
+The accumulation function is a special case of the amount function where :math:`K=1`:
+
+.. math::
+   a(t) = A_1(t)
+
+It is often convenient to use this form to explore the growth of money without having to bother with the principal.
+
+The amount and accumulation functions are often related by the following expression:
+
+.. math::
+   A_K(t) = Ka(t)
+
+
+Examples
+========================
+
+TmVal's **Accumulation** class models accumulation functions.
+
+Suppose money exhibits a quadratic growth pattern, specified by the amount function:
+
+.. math::
+   a(t) = .05t^2 + .05t + 1
+
+How much does $1 invested at time 0 grow to at time 5? To solve this problem, we import the Accumulation class, supply the growth function in a similar manner as we had done with the Amount class, except we do not need to supply a value for :math:`K`.
+
+.. ipython:: python
+
+   from tmval import Accumulation
+
+   def f(t):
+       return .05 * (t **2) + .05 * t + 1
+
+   my_acc = Accumulation(f=f)
+
+    print(my_acc.val(5))
+
+Note that we could have also solved this problem with the Amount class, by setting :math:`K=1`.
+
+.. ipython:: python
+
+   from tmval import Amount
+
+   def f(t, k):
+       return k * (.05 * (t **2) + .05 * t + 1)
+
+   my_amt = Amount(f=f, k=1)
+
+   print(my_amt.val(5))
+
+If the amount and accumulation functions are proportionally related, we can extract the accumulation function from the Amount class by calling the **get_accumulation()** method, which returns an Accumulation class derived from the Amount class:
+
+.. ipython:: python
+
+   from tmval import Amount
+
+   def f(t, k):
+       return k * (.05 * (t **2) + .05 * t + 1)
+
+   my_amt = Amount(f=f, k=1)
+
+   my_acc = my_amt.get_accumulation()
+
+   print(my_acc.val(5))
+
+