Calculus 1

Calculus 1 equations

Models and functions Limits and derivatives Applying derivatives Integral theory

  Models and functions  

Definition of a function

A function is an action that takes on element in a set A and assigns it to one and only one element in a set B.

Vertical line test

Take a graph on a 2D plane, and draw a vertical line. If it intersects the graph only once, it is a function.

Piecewise function

$$ \lvert c \rvert = c \quad \textnormal{if} \quad c \geq 0 $$ $$ \lvert c \rvert = - c \quad \textnormal{if} \quad c \leq 0 $$

Increasing and decreasing functions

A function is increasing on an interval I if \begin{equation} f_{x_1} < f_{x_2} \quad \textnormal{whenever} \quad x_1 < x_2 \quad \textnormal{ on } I \end{equation}

  Limits and derivatives  

Non-rigorous definition of a limit

Say that f(x) is defined near the point x = a, then we say that

\begin{equation} \lim_{x \to a} f(x) = C , \qquad C \quad \textnormal{ is the limit} \end{equation}

Limit is like an approximation.

One-sided limit

When we approach x = a from either less than a or greater than a, we write

\begin{equation} \lim_{x \to a^-} f(x) = C , \qquad \lim_{x \to a^+} f(x) = C \qquad \textnormal{One sided limit} \end{equation}

And this is to say

\begin{equation} \lim_{x \to a} f(x) = C \quad \textnormal{if and only if} \quad \lim_{x \to a^-} f(x) = C \quad \textnormal{and} \quad \lim_{x \to a^+} f(x) = C \end{equation}

Non-rigorous infinite limit

Say that f(x) is defined on both sides of x = a, but not necessarily at a itself. If we can make f(x) arbitrarily large by making x closer and closer to a, such as f(a - 1), f(a - .5), f(a - 0.1), and f(a + 1), f(a + .5), f(a + 0.1), then we say

\begin{equation} \lim_{x \to a} f(x) = \infty , \qquad \textnormal{Infinite limit} \end{equation}

The function is approximately going to... ∞

Similarly, if we can make f(x) arbitrarily small in the negative direction by making x closer and closer to a, such as f(a - 1), f(a - .5), f(a - 0.1), and f(a + 1), f(a + .5), f(a + 0.1), then we say

\begin{equation} \lim_{x \to a} f(x) = - \infty , \end{equation}

Vertical asymptote

A function has a vertical asymptote at x = a if one of these statements is true

\begin{equation} \lim_{x \to a} f(x) = \infty \quad \lim_{x \to a^-} f(x) = \infty \quad \lim_{x \to a^+} f(x) = \infty \end{equation} \begin{equation} \lim_{x \to a} f(x) = - \infty \quad \lim_{x \to a^-} f(x) = - \infty \quad \lim_{x \to a^+} f(x) = - \infty \end{equation}

Using limit laws

Suppose we have two functions on a graph, f(x) and h(x), and the limits

\begin{equation} \lim_{x \to a} f(x) \quad \textnormal{and} \quad \lim_{x \to a} h(x) \end{equation}

exist. This means that we can make an approximation at x = a for both of them (the function is tending towards some defined number), then


\begin{equation} 1. \quad \lim_{x \to a} \, [f(x) + h(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} h(x) \end{equation}
\begin{equation} 2. \quad \lim_{x \to a} \, [f(x) - h(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} h(x) \end{equation}
\begin{equation} 3. \quad \lim_{x \to a} \, [c \, f(x)] = c \, \lim_{x \to a} f(x) \end{equation}
\begin{equation} 4. \quad \lim_{x \to a} \, [f(x) \, h(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} h(x) \end{equation}
\begin{equation} 5. \quad \lim_{x \to a} \frac{f(x)}{h(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} h(x)} \quad \textnormal{if} \quad \lim_{x \to a} h(x) \neq 0 \end{equation}

Power law for limits

\begin{equation} 6. \quad \lim_{x \to a} \, [f(x)]^n = [ \lim_{x \to a} f(x)]^n \end{equation}

Limit of a constant

\begin{equation} 7. \quad \lim_{x \to a} c = c \end{equation} \begin{equation} 8. \quad \lim_{x \to a} x = a \end{equation} \begin{equation} 9. \quad \lim_{x \to a} x^n = a^n \qquad \textnormal{with } n \textnormal{ a positive integer} \end{equation} \begin{equation} 10. \quad \lim_{x \to a} \sqrt[n]{x} = \sqrt[n]{a} \end{equation} \begin{equation} 11. \quad \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)} \end{equation}

  Applying derivatives  

  Integral theory  

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Taken from Stewart calculus.