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 ith subinterval then the definite integral of \(f\) from \(a\) to \(b\) follows:
\[\int_a^b {f\left( x \right)dx} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {f\left( {x_i^*} \right)} \,\Delta x\]
Derivative of the Sine Function: \[\frac{d}{{dx}}\left( {\cos x} \right) = - \sin x\]
Derivative of the Sine Function: \[\frac{d}{{dx}}\left( {\sin x} \right) = \cos x\]
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.
Derivative of a Function at \(a\), if it exists:
\[f'\left( a \right) = \mathop {\lim }\limits_{h \to 0} \frac{{f\left( {a + h} \right) - f\left( a \right)}}{h}\]
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.
