Calculus for Policy Students: Don’t Panic
Calculus Introduction / Review
Don’t worry, you won’t need to actually use much calculus
- But, understand the concepts of limits, derivatives and integrals are important
Limits are crucial for understanding both mechanics of hypothesis testing, and for understanding derivatives
Derivatives are important for understanding instantaneous rates of changes and for optimization
Integrals are crucial for hypothesis testing
Limits
What is a Limit?
A limit helps us understand the behavior of a function as it approaches a certain point.
It tells us the value that a function gets close to as the input approaches a specific value, even if it doesn’t actually reach that value.
Mathematically, we write:
\[ \lim_{{x \to c}} f(x) = L \]
This means that as x gets closer to c, the value of f(x) approaches L.
Formal Definition of a Limit
For a function \(f(x)\), the statement:
\[ \lim_{{x \to c}} f(x) = L \]
means that for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that:
\[ 0 < |x - c| < \delta \implies |f(x) - L| < \epsilon \]
Breaking Down the Definition
If \(\lim \limits_{x \to c} f(x) = L\)
\[ 0 < |x - c| < \delta \implies |f(x) - L| < \epsilon \]
\(\epsilon\) represents how close we want \(f(x)\) to be to L.
\(\delta\) represents how close x must be to c to guarantee that f(x) is within \(\epsilon\) of L
The limit exists if we can find such a \(\delta\) for any \(\epsilon\)
TL;DR: As x gets very close to x, f(x) gets arbitrarily close to L.
Solving Limits
1. Direct Substitution
If ( f(x) ) is continuous at ( x = c ), you can find the limit by substituting ( c ) directly into the function: \[ \lim_{{x \to c}} f(x) = f(c) \]
2. Factoring
If direct substitution leads to an indeterminate form like \(\frac{0}{0}\), try factoring the numerator and denominator, and then simplify: \[ \lim_{{x \to c}} \frac{{x^2 - 1}}{{x - 1}} = \lim_{{x \to c}} \frac{{(x - 1)(x + 1)}}{{x - 1}} = \lim_{{x \to c}} (x + 1) \]
Solving Limits
3. L’Hôpital’s Rule
If direct substitution leads to an indeterminate form, differentiate the numerator and denominator: \[ \lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} \]
4. Limits at Infinity
Divide by the highest power of x to solve limits as \(x \to \infty\): \[ \lim_{{x \to \infty}} \frac{x + 1}{x} = \lim_{{x \to \infty}} \left( 1 + \frac{1}{x} \right) = 1 \]
Example: Limit of a Function
Consider the function:
\[ f(x) = \frac{{x^2 - 1}}{{x - 1}} \]
If we plug in x = 1, we get \(\frac{0}{0}\), but the limit exists!
\[ \frac{{x^2 - 1}}{{x - 1}} = \frac{{(x - 1)(x + 1)}}{{x - 1}} = x + 1 \]
Thus, \(\lim\limits_{x\to 1} f(x) = 2\)
Visualizing Limits
Visualizing Limits
\[ f(x) = \frac{x+1}{x} \]
Derivatives
Understanding Derivatives
A derivative represents the rate at which a function is changing at any given point. It’s essentially the slope of the function at that point.
The derivative of \(f(x)\) with respect to \(x\) is defined as:
\[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \]
This formula calculates the instantaneous rate of change of \(f(x)\) as \(\Delta x\) becomes infinitesimally small.
Visualizing Derivatives
Notice how as \(\Delta x\) gets smaller, we approach the true slope (represented by the thick red line).
Rules for Differentiation
Polynomial Rule
The Rule
For a polynomial function of the form:
\[ f(x) = x^n \]
The derivative with respect to x is:
\[ f'(x) = n x^{n - 1} \]
Chain Rule
The Rule
For a composite function f(x) = g(h(x)) , the derivative with respect to x is:
\[f'(x) = g'(h(x)) \cdot h'(x) \]
Product Rule
The Rule
For two differentiable functions u(x) and v(x):
\[ \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) v(x) + u(x) v'(x) \]
Integrals
Understanding Integrals
An integral represents the accumulation of quantities, such as areas under curves. It’s essentially the inverse operation of differentiation.
The definite integral of \(f(x)\) from \(a\) to \(b\) is written as:
\[ \int_{a}^{b} f(x) \, dx \]
This computes the total accumulation of \(f(x)\) between \(x = a\) and \(x = b\).
Area Under a Curve
To find the area under a curve \(f(x)\) from a to b , we can:
- Partition the interval [a, b] into small subintervals.
- Approximate the area under the curve in each subinterval using rectangles.
- Sum the areas of all rectangles.
- Take the limit as the number of rectangles approaches infinity.
Approximating area under curve using 10 rectangles
Approximating the Area Under a Curve Using 10 Rectangles
100 Rectangles
Approximating the Area Under a Curve Using 100 Rectangles
1000 Rectangles
Approximating the Area Under a Curve Using 1000 Rectangles
Antiderivatives
An antiderivative of a function f(x) is another function F(x) such that:
\[ F'(x) = f(x) \]
In other words, the derivative of F(x) is f(x) .
Antiderivatives are also known as indefinite integrals.
Notation for Antiderivatives
The antiderivative of ( f(x) ) is denoted by:
\[ \int f(x) \, dx = F(x) + C \]
\(\int\) is the integral symbol.
C is the constant of integration, representing all possible antiderivatives.
- This is because we can only recover the original function up to a constant, as the derivative of a constant is 0.
Example of an Antiderivative
Find the antiderivative of f(x) = 2x.
Solution:
We are looking for a function F(x) such that F’(x) = 2x.
The antiderivative is:
\[ \int 2x \, dx = x^2 + C \]
Understanding Antiderivatives
Antidifferentiation is the reverse process of differentiation.
Antiderivatives allow us to recover the original function from its derivative.
They are essential in computing definite integrals using the Fundamental Theorem of Calculus.
Fundamental Theorem of Calculus
If f is continuous on [a, b] and F is an antiderivative of f, then:
\[\int_{a}^{b} f(x) dx = F(b) - F(a) \]
Applying the Fundamental Theorem
Compute the area under \(f(x) = x^{2}\) from x = 0 to x = 3.
Solution
Find the antiderivative F(x) :
\[F(x) = \int f(x) dx = \int x^2 dx = \frac{x^3}{3} + C\]
Apply the Fundamental Theorem:
\[\int_{0}^{3} x^2 dx = F(3) - F(0) = \left( \frac{3^3}{3} \right) - \left( \frac{0^3}{3} \right) = \left( \frac{27}{3} \right) - 0 = 9 \]
Example: Total Accumulation
Imagine you want to model the total tax revenue of a city over the next 5 years.
We would expect revenue to increase as a function of economic growth, inflation and population growth.
Let’s assume that the town has current tax revenues of $10,000,000, and that revenue grows at 7% per year.
Exponential Growth Model
The exponential growth formula is
\[ R(t)=R(0) ⋅e^{kt} \]where R(0) is revenue at time t = 0, k is the growth rate and t is the time that has passed.
\[ \text{Total Revenue} = \int_{0}^{5} R(t) dt = \int_{0}^{5} 10,000,000\times e^{.07t}dt \]
Solve the Integral
We pull the initial revenue out front since it’s a constant which leaves us
\[ \text{Total Revenue} = 10,000,000 \int_{0}^{5}e^{.007t}dt \]
The integral of \(e^{kt}\) with respect to t is \[\frac{1}{K} e^{kt}\]
Use the FTC
\[ \text{Total Revenue} = 10,000,000 \times(\frac{e^{.35} - e^{0}}{0.07}) \]
Which ends up coming out to a Total Revenue of
\[ 59,860,000 \]
Why We Care/What’s Next
There are many cases where we want to know to accumulation of some quantity (economic growth, expenses, etc) over time.
In this course, understanding integration and the area under the curve will be crucial to understanding hypothesis testing