**Interactive Instructions:**Move any of the green points. Also, you can change the orientation below.

**NOTES**: Hyperbolas ]

Showing posts with label **interactive**. Show all posts

Showing posts with label **interactive**. Show all posts

Hyperbola:
Horizontal
Vertical

[ **NOTES**: Hyperbolas ]

**Mean Value Theorem for Integrals:** If \(f\) is continuous on \([a,b]\), then there exists a number \(c\) in \([a,b]\) such that \[\int_a^b {f\left( x \right)dx = f\left( c \right)\left( {b - a} \right)} \]

**Net Area:** If the function *f* takes on both positive and negative values, then the Riemann sum is the result of the area under the curve above the *x*-axis less the area under the *x*-axis above the curve.

**Definite Integral:** If \(f\) is a continuous function defined on the interval \([a, b]\) that is divided into \(n\) equal subintervals \(\Delta x = \frac{{b - a}}{n}\) where \(x_i^*\) lies in the *i*th subinterval then the definite integral of \(f\) from \(a\) to \(b\) follows:

\[\int_a^b {f\left( x \right)dx} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {f\left( {x_i^*} \right)} \,\Delta x\]

Riemann Sum:
Left
Middle
Right
Trapezoidal

**Fermat's Theorem:** If \(f\) is a local maximum or minimum at \(c\), and if \(f'(c)\) exists, then \(f'(c)=0\).

**Critical Number:** A number \(c\) in the domain of \(f\) such that either \(f'(c)=0\) or \(f'(c)\) does not exist.

- \(f\) is continuous on a closed interval \([a, b]\).
- \(f\) is differentiable on the open interval \((a,b)\).

\[f'\left( p \right) = \frac{{f\left( b \right) - f\left( a \right)}}{{b - a}}\]

**Definition:** The tangent line to the curve \(y = f(x)\) at the point \(P(a, f(a))\) is the line through \(P\) with slope

\[m = \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right) - f\left( a \right)}}{{x - a}}\]

provided that this limit exists.

[ **NOTES**: The Circumcenter of a Triangle ]

[ **NOTES**: Circles ]

[ **NOTES**: Perpendicular Lines ]

[ **NOTES**: Parallel Lines ]

Labels:
algebra,
interactive,
line,
lines,
math,
parallel,
same slope,
slope

[ **NOTES**: Slope and y-Intercept ]

[ **NOTES**: Rectangular Coordinate System ]

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