## Calculus Review |

You should revise the limit definitions of the derivative and the Riemann integral of a function from a standard text book. The following are some theorems which will be used later.

**Theorem.**
Let \(f\) be a ** differentiable** function on \([a,b]\).

- \(f\) is increasing on \([a,b]\) if and only if \(f'(x)>0\) for all \(x\in [a,b]\).
- If \(f\) has a local maximum or minimum value at \(c\), then \(f'(c)=0\) (\(c\) is a critical number).
- If \(f'(c)=0\) and \(f''(c)<0\), then \(f(c)\) is a local maximum value.
- If \(f'(c)=0\) and \(f''(c)>0\), then \(f(c)\) is a local minimum value.

**Theorem.** Let \(f\) be a ** continuous** function on \([a,b]\). If \(f(c)\) is the absolute maximum or minimum value
of \(f\) on \([a,b]\), then either \(f'(c)\) does not exist or \(f'(c)=0\) or \(c=a\) or \(c=b\).

- \(f\) is continuous on \([a,b]\), and
- \(N\) is a number between \(f(a)\) and \(f(b)\).

In the particular case when \(f(a)f(b)<0\), i.e., \(f(a)\) and \(f(b)\) are of opposite signs, there is at least one root \(c\) of \(f\) in \((a,b)\).

- \(f\) is continuous on \([a,b]\), and
- \(f\) is differentiable on \((a,b)\).

- \(f^{(n)}\) is continuous on \([a,b]\), and
- \(f^{(n)}\) is differentiable on \((a,b)\).

Sometimes we simply write \(f(x)=T_n(x)+R_n(x)\), where \(T_n(x)=\displaystyle\sum_{k=0}^n \frac{f^{(k)}(c)}{k!}(x-c)^k\)
is the ** Taylor polynomial of degree \(n\)** and \(R_n(x)=\displaystyle\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-c)^{n+1}\)
is the ** remainder term**.

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