31BMidterm2Study Guide - 31B Notes Sudesh Kalyanswamy 1...

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31B Notes Sudesh Kalyanswamy 1 Exponential Functions (7.1) The following theorem pretty much summarizes section 7.1. Theorem 1.1. Rules for exponentials: (1) Exponential Rules (Algebra): (a) a x · a y = a x + y (b) a x a y = a x - y (c) ( a x ) y = a xy (2) Derivatives of Exponentials: (a) f ( x ) = e x = f 0 ( x ) = e x (b) f ( x ) = a x = f 0 ( x ) = a x ln( a ) Example 1.2. Simplify 10 2 (2 - 2 + 5 - 2 )
Example 1.3. Simplify (9 - 1 / 6 ) 3 .
Example 1.4. Solve 2 x +1 = 4 .
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Solution. To solve this, we want to get the same base, so write 25 = 5 2 . Therefore 5 x = 25 1 - x = 5 x = (5 2 ) 1 - x = 5 x = 5 2(1 - x ) . Now set the exponents equal: x = 2(1 - x ) = x = 2 - 2 x = 3 x = 2 = x = 2 3 . Example 1.6. What is the tangent line to f ( x ) = e sin( x ) at x = 0 ?
Example 1.7. Find the derivative of f ( x ) = xe - 2 x +3 .
Example 1.8. Find the derivative of f ( x ) = e x +1 (2 x +1) 3 .
Example 1.9. Show f ( x ) = x + e 2 x is always increasing.
Example 1.12. Evaluate R x e x 2 dx .

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