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1(x) = x2 is an antiderivative of f(x) = 2x. F 2(x) = x2 +2004 is also an antiderivative of f(x) = 2x. G(t) = 1 2 sin(2t +1) is an antiderivative of g(t) = cos(2t+1). The Fundamental Theorem of Calculus states that if a function y = f(x) is continuous on an interval a ≤ x ≤ b, then there always exists an antiderivative F(x) of f, and one ...
Chapter 1 Essentials of Fractional Calculus In this chapter we introduce the linear operators of fractional in-tegration and fractional di erentiation in the framework of the so-called fractional calculus. Our approach is essentially based on an integral formulation of the fractional calculus acting on su ciently
4.1 Trigonometric identities Euler’s formula allows one to derive the non-trivial trigonometric identities quite simply from the properties of the exponential. For example, the addition for-mulas can be found as follows: cos( 1 + 2) =Re(ei( 1+ 2)) =Re(ei 1ei 2) =Re((cos 1 + isin 1)(cos 2 + isin 2)) =cos 1 cos 2 sin 1 sin 2 and sin( 1 + 2) =Im ...
Evaluate f ( x ) at all points found in Step 1. minimum, or neither if f ¢ ¢ ( c ) = 0 . Evaluate f ( a ) and f ( b ) . Identify the abs. max. (largest function value) and the abs. min.(smallest function value) from the evaluations in Steps 2 & 3.
know how cos, sin and tan functions are defined for all real numbers; be able to sketch the graph of certain trigonometric functions; know how to differentiate the cos, sin and tan functions; understand the definition of the inverse function f−1(x) = cos− 1(x).
The sine, cosine and tangent of an angle are all defined in terms of trigonometry, but they can also be expressed as functions. In this unit we examine these functions and their graphs. We also see how to restrict the domain of each function in order to define an inverse function.
One is a simple formula, and the other is much more complicated because of the multivalued nature of the inverse function: sinIsin -1 HzLM−zsin -1 HsinHzLL−z’;- p
