We go over some textbook exercises for exponential functions.

## Franchise Expansion

A franchise business is selling franchises for pizza shops throughout the country. The marketing manager estimates that the number of franchises, $$N$$, will increase at the rate of 10% per year, that is,

$\frac{dN}{dt} = 0.10N$

1. Find the function that satisfies this equation, assuming that the number of franchises at $$t = 0$$ is 50.
2. How many franchises will be there in 20 years?
3. In what period of time will the initial number of 50 franchises double?
from sympy import *
from math import e
from sympy.abc import N, t

N = Function("N")(t)
dNdt = Eq(N.diff(t), 0.10 * N)
f = dsolve(dNdt, N)
f
## Eq(N(t), C1*exp(0.1*t))
def f(t, c):
return c * e**(0.1 * t)

f(20, 50)
## 369.45280494653247
solve(50 * e**(0.1 * t) - 100, t)
## [6.93147180559944]

## Compound Interest

Suppose that $$P_0$$ is invested in a fund for which interest is compounded continuously at 5.9% per year. That is, the balance $$P$$ grows at the rate given by

$\frac{dP}{dt} = 0.059P$

1. Find the function that satisfies the equation.
2. Suppose that $1000 is invested, what is the balance after 1 year and 2 years? 3. When will a$1000 investment double itself?
from sympy.abc import P

P = Function("P")(t)
dPdt = Eq(P.diff(t), 0.059 * P)
dPdt
## Eq(Derivative(P(t), t), 0.059*P(t))
dsolve(dPdt, P)
## Eq(P(t), C1*exp(0.059*t))
def f(t, c):
return c * e**(0.059 * t)

f(1, 1000)
## 1060.775240740159
f(2, 1000)
## 1125.2441113673424
solve(1000 * e**(0.059 * t) - 2000, t)
## [11.7482572976262]

Peter Minuit o the Dutch West India Company purchased Manhattan Island from the natives living there in 1626 for $24 worth of merchandise. Assuming an exponential rate of inflation of 5%, how much will Manhattan be worth in 2020? Let’s assume the growth model for the value of the island is given by $\frac{dV}{dt} = 0.05V$ The function that satisfies this rate of change is $V(t) = ce^{0.05t}$ from sympy.abc import V V = Function("V")(t) dVdt = Eq(V.diff(t), 0.05 * V) dVdt ## Eq(Derivative(V(t), t), 0.05*V(t)) f = dsolve(dVdt, V) f ## Eq(V(t), C1*exp(0.05*t)) years = 2020 - 1626 def f(t, c): return c * e**(0.05*t) f(years, 24) ## 8626061203.204306 ## Cost of Hershey Bar The cost of a Hershey bar was$0.05 in 1962 and $0.75 in 2010. 1. Find an exponential function that fits the data. 2. Predict the cost of a Hershey bar in 2015 and 2025. The assumtion is that the value of a bar has grown exponentially $\frac{dV}{dt} = kV$ Because the value at $$t = 0$$ is$0.05, we can say that the exponential function is

$V(t) = 0.05e^{kt}$

We know that 48 years after 1962, the value is $0.75, therefire $0.75 = 0.05e^{k(48)}$ and we need to find $$k$$. from sympy.abc import k years = 2010 - 1962 equation = Eq(0.75, 0.05 * e**(k*years)) k = solve(equation, k) k  ## 0.0564177125229626 def f(t, c): return 0.05 * e**(c*t) f(2015 - 1962, k) ## 0.994422104712232 f(2025 - 1962, k) ## 1.74819462993714 ## Diffusion of Information Pharmaceutical firms invest significantly in testing new medications. After a drug is approved by the FDA, it still takes time for physicians to fully accept and start prescribing the medication. The acceptance by physicians approaches a limit value of 100%, or 1, after time $$t$$, in months. Suppose that the percentage $$P$$ of physicians prescribing a new cancer medication after $$t$$ months is approximated by $P(t) = 100(1 - e^{-0.4t})$ 1. What percentage of doctors are prescribing the medication after 0, 1, 2, 3, 5, 12, and 16 months? 2. Find $$P^\prime(7)$$ and interpret its meaning. 3. Sketch a graph of the function. def p(t): return 100 * (1 - e**(-0.4 *t)) percentages = [round(p(t), 2) for t in [1, 2, 3, 5, 12, 16]] percentages ## [32.97, 55.07, 69.88, 86.47, 99.18, 99.83] expr = 100 * (1 - e**(-0.4 *t)) expr.diff() ## 40.0*2.71828182845905**(-0.4*t) The equation $p^\prime(t) = 40e^{-0.4t}$ tells us the rate of change of the percentage of phyicians that accept the new medication at any given time (in months). import numpy as np import matplotlib.pyplot as plt x_values = np.linspace(0, 20) plt.plot(x_values, p(x_values)) plt.title("% physicians accepting and prescribing new medication") ## Hullian Learning Model The Hullian learning model asserts that the probability $$p$$ of mastering a task after $$t$$ learning trials is approximated by $p(t) = 1 - e^{-kt}$ where $$k$$ is a constant that depends on the task to be learned. Suppose a new dance is taught to an aerobics class. For this particular dance, the constant $$k = 0.28$$. 1. What is the probability of mastering the dance’s steps in 1, 2, 5, 11, 16, 20 and trials? 2. Find the rate of change, $$p^\prime(t)$$. 3. Sketch a graph of the function. def p(t, k): return 1 - e**(-k*t) constant = 0.28 probabilities = [round(p(t, constant), 2) for t in [1, 2, 5, 11, 16, 20]] probabilities ## [0.24, 0.43, 0.75, 0.95, 0.99, 1.0] expr = 1 - e**(-k*t) expr.diff() ## 0.0564177125229626*2.71828182845905**(-0.0564177125229626*t) x_values = np.linspace(0, 25) plt.plot(x_values, p(x_values, constant)) plt.title("$p(t)$of mastering dance steps after$t\$ trials") 