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Let $f(x)$ be a real-valued function on $[a,b]$. To give a Riemann sum of $f$ on $[a,b]$, we follow two steps: Give a partition $a=x_0<x_1<\cdots<x_n=b$ of $[a,b]$. Choose sample points $x_i^*$ in each subinterval $[x_{i},x_{i+1}]$ for $i=0,1,\ldots, n-1$. Then, a Riemann sum is \(f(x_0^\*)(x_1-x_0)+f(x_1^\*)(x_2-x_1)+\cdots+f(x_{n-1}^\*)(x_{n}-x_{n-1})=\sum_{i=0}^{n-1}f(x_i^\*)(x_i-x_{i-1}).\) [Read More]

We say a limit $\displaystyle\lim_{x\to a}\frac{f(x)}{g(x)}$ is called an indeterminate form if the fraction $\displaystyle\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}$ is of one of the following form \(\frac{0}{0},\text{or } \pm\frac{\infty}{\infty}.\) [Read More]

%%{init: {"flowchart":{"useMaxWidth": 0}}}%% flowchart LR; Limit[<a href='../books/calculus/1-definition-of-limits'>Definition of Limit</a>]; LimitLaw[<a href='../books/calculus/1-1-how-to-evaluate-limits#limit-laws'>Limit laws</a>]; LimitTech[<a href='../books/calculus/1-1-how-to-evaluate-limits#algebraic-tricks'>Useful algebraic techniques of evaluating limits</a>]; Squeeze[<a href='../books/calculus/1-1-how-to-evaluate-limits#the-squeeze-theorem'>The squeeze theorem</a>]; LHopital[<a href='../books/calculus/1-1-how-to-evaluate-limits#lhpitals-rule'>L'Hopital Rule</a>]; Asymptote[<a href='../books/calculus/1-2-asymptotes'>Vertical asymptote Horizontal asymptote Slant asymptote</a>]; Continuity[<a href='../books/calculus/2-definition-of-continuity'>Definition of continuity</a>]; EVT[<a href='../books/calculus/2-1-theorems-assuming-continuity#extreme-value-theorem'>Extreme value theorem</a>]; IVT[<a href='../books/calculus/2-1-theorems-assuming-continuity#intermediate-value-theorem'>Intermediate value theorem</a>]; Diff[<a href='../books/calculus/3-definition-of-derivatives'>Definition of derivative</a>]; BasicFunction[<a... [Read More]

In calculus, the racetrack principle describes the behavior and development of two functions based on their derivatives. This principle originates from a straightforward concept: if I run faster than you, I should reach the finish line sooner. [Read More]

What does derivatives tell us about the graph of a function (continuous) [Read More]

Mean Value Theorem [Read More]

All works must be seen! [Read More]