Calculus Complexities - Intermediate

Calculus is the mathematical study of change, through the ideas of infinite series. It is developed by working with the infinitesimal, which is a really really small quantity. From there, we develop the idea of a Epsilon-Delta Definition of a Limit, which allows us to rigorously explain why a sequence or function tends to a particular unique value.

Here are some tips to get started:

1. Understand how to find the limit of a sequence.
2. Review the derivatives and integrals of basic functions.
3. Remember how to derive the most beautiful math equation \( e^{i \pi } + 1 = 0 \).
4. Understand the fundamental theorem of calculus.
5. If you are stuck, read the solutions to grasp these concepts better.

True or False?

\(i^ i \) is not a real number.

The statement is false. As it turns out, the value is real.

Recall that \( e ^ { i \pi } = -1 \). Substituting \( e ^ { i \pi / 2 } = i \) for the base, we obtain

\[ i ^ i = \left( e ^ { i \pi / 2 } \right) ^ i = e^{ i i \pi / 2 } = e ^ { - \pi / 2 } \approx 0.2078. \]

Note: Since we are working complex exponentiation, there are actually multiple values that \(i^i \) can take. In a similar manner to the above derivation, these values are all of the form \( e ^ { - ( 2k+1) \pi / 2 } \). Likewise, these are real valued, and are approximately \( 0.2078 \times 2^{k \pi } \).

Calculus topics can be broadly classified as

• Differential calculus studies the Derivative of a function, which is defined through first principles. It measures the instantaneous rate of change of the function, which is obtained by taking the limit of the secant as it gets closer to the point. It is concerned with Slope of a Curve, Related Rates of Change and Local Linear Approximation.
  
• Integral calculus studies the Anti-derivative of a function, which is defined as the inverse function to the derivative. The definite integral gives us the limit of the Riemann Sums, to find the area under a curve. It is concerned with Area Between Curves and Volume of Revolution.

These concepts are related via the Fundamental Theorem of Calculus, which states that

If \( f\) is a continuous function on the interval \( [ a, b] \), and \(F \) is a function whose derivative is \(f\) on the interval \( (a, b) \), then we have

\[\begin{array} &\int_a^b f(t) \, dt = F(b) - F(a) &\text{ and } &\frac{ d}{dx} \int_a ^ x f(t) \, dt = f(x) \text{ for } x \in (a, b). \ _\square \end{array}\]