Calculus Complexities - Basic

When we shoot an arrow, how fast does it travel? If we look at an instantaneous point in time, the arrow covers no distance over no time, and thus its velocity is \( \frac{0}{0},\) which takes on an indeterminate form!

Questions like this plagued the ancients, which led to Xeno's paradox about how the legitimacy of splitting up the whole into parts. Calculus was developed to help us construct quantitative models of change, and to draw conclusions from the results. By expressing the effects of changing conditions, we can deduce how the system will be changing over time, and thus give us great control over these processes. The fundamental idea of calculus is to study how things change instantaneously (over very small intervals of time), which allows us to model the velocity of an arrow through the relation to its position.

Here are some tips to get started:

1. Be familiar with sequences.
2. Listing out initial terms to guess the limit of a sequence.
3. Review the derivatives and integrals of basic functions.
4. Question your assumptions! Somethings, strange things can happen.
5. Understand the fundamental theorem of calculus.
6. If you are stuck, read the solutions to grasp these concepts better.

True or False?

\[ e ^ { i \pi} + 1 = 0. \]

This identity is true. It is known as Euler's identity, which is a beautiful mathematical equation.

One way of deriving it is to use Euler's Formula, which states that

\[ e ^ { i \theta } = \cos \theta + i \sin \theta .\]

Then, since \( \cos \pi = -1 \) and \( \sin \pi = 0 \), substituting in \( \theta = \pi \) gives

\[ e ^ { i \pi } = -1 + i \times 0 = -1. \ _\square\]

Back to Quiz: Calculus Complexities - Basic

To get started in calculus, check out the following:

• Sequences and Series: Be familiar with finding the sum of an infinite geometric progression, and finding the limit of a sequence.
• Limit of functions: Be comfortable with various approaches like substitution, factorization and rationalization.
• Infinitesimal changes: This explains how we can use the idea of determining the instantaneous rate of change as the limit of the average rate of change. In this way, the arrow's velocity is not \( \frac{0}{0} \).