Exponential Function Context and Data Modeling

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

Exponential modeling shows up most often in three contexts: doubling time / half-life, compound interest, and biological populations. The model is always $P(t) = P_0 \cdot b^t$ or its continuous form $P(t) = P_0 \cdot e^{kt}$ . Doubling time satisfies $b^{t_d} = 2$ ; half-life satisfies $b^{t_h} = 1/2$ . Both reduce to logarithm computations.

Exponential models

Discrete

P(t) = P_0 \cdot b^t

Continuous

P(t) = P_0 \cdot e^{kt}

Doubling time and half-life

Doubling time

t_{\text{double}} = \frac{\ln 2}{\ln b} = \frac{\ln 2}{k}

Half-life

t_{1/2} = \frac{\ln 2}{|\ln b|} = \frac{\ln 2}{|k|}

Compound interest

Periodic

A = P \left(1 + \frac{r}{n}\right)^{n t}

Continuous

A = P e^{r t}

Caution: 'Compounded quarterly' means

n = 4

per year, not 'compound 4 times'. The total exponent is

n \cdot t

Type 1

Doubling time and half-life

Plug into the doubling-time or half-life formula. Both reduce to $\ln 2$ divided by the appropriate growth or decay rate.

Example 1

A population grows by 10% per year. What is the doubling time? (A) About 5 years (B) About 7.3 years (C) About 10 years (D) About 14 years

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Type 2

Compound interest models

Identify $P$ , $r$ , $n$ , and $t$ . Plug into the periodic formula or use $A = Pe^{rt}$ for continuous.

Example 2

$2,000 invested at 6% compounded quarterly. Find the balance after 5 years. (A) About $2,600 (B) About $2,694 (C) About $2,800 (D) About $3,000

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Type 3

Build an exponential model from two data points

Divide the second equation by the first to isolate $b^{\Delta t}$ . Take the appropriate root for $b$ , then back out the initial value $a$ .

Example 3

An exponential model has

P(2) = 50

and

P(5) = 200

. Approximately, what are

a

and

b

P(t) = a \cdot b^t

? (A)

a \approx 50

b \approx 2

(B)

a \approx 19.8

b \approx 1.59

(C)

a \approx 200

b \approx 1

(D)

a \approx 30

b \approx 1.33

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