Calculus and Vectors - How to get an A+ B Link between a function and its derivative Ex 6. The graphs of a function and its first and second Consider a double differentiable function derivatives are represented on the same grid. Identify y = f(x) ( f' (x) and f"(x) exist). Then: each of them. 1. f'(x) is the slope of the tangent at P(x, f(x)) 2. If f' (x) = 0, then P(x, f(x)) is a local extrema and tangent is horizontal. 3. If f'(x) > 0, then the function y = f(x) is increasing. 4. If f' (x) 0, then f'(x) is increasing and y = f(x) is concave upward. 7. If f"(x) 0,b=1 evaluate each limit. a) lim 2. below: The graph of the exponential function is represented y = b* ; b 1 b) lim 2. C) lim 0.2. d) lim 0.2. The x-axis ( y = 0 ) is a horizontal asymptote. B Number e Ex 2. Estimate the number e using formula (1) and by The number e is defined by: taking n = 100000 . e - lim 1+1) (1 ) which can be written also as: e = lim(1+ u ) 4 ( 2 ) C Exponential Function Ex 3. Estimate ve using formula (2) and by taking The exponential function e* may be evaluate using n =100000 the limit: ex = lim 1+ (3) Proof : lim 1+ * - lim (1 + * )* = 1im ( 1 + * ) 3 = lim ( 1 + u ) i D Derivative of e.T Ex 4. Differentiate and simplify. ( e * )' = ex a) ret dex = ex ( 4 ) der Proof: b) e.x / 2 ( 1 + * ) - ex = n ( It * ) ( 4 ) - (2 5 ) 5.1 5.2 Derivative of Exponential Function @2010 lulia & Teodoru Gugoiu - Page 1 of 3Calculus and Vectors - How to get an A+ E Derivative of ef(.x) Ex 5. Differentiate. Using (4) and the chain rule: a ) e-3.x ( es (. ) ) ' = ef (* ) fi (x ) def ( = ) = efl * ) fi (x ) (5 ) b) e-1!x' dx c ) ever '+ 1 Ex 6. The hyperbolic functions are defined by: c) (cosh .x)'= sinh x sinh r = ex - er cosh x =. -, tanh x = sinh x 2 2 cosh x Prove that: a) cosh- x - sinh- x = 1 1 d) (tanh x)'= cosh x b) (sinh x)'= cosh x F Derivative of b*, b> 0,b #1 Ex 7.. Differentiate. ( 6 * ) ' = ( Inb ) 6-* a) 3. aft = ( Inb ) 6 * (6) dr Proof: b ) x 2 2 .* (6* )' = (e*lnb)' = elnb ( Inb) = (Inb)6* c ) (4 * + xx + ) 3 G Derivative of bf(.x) Ex 8. Differentiate. Using (6) and the chain rule: a ) 2 - 1 ( 6 f (.) ) ' = ( Inb ) b f ( ) f' (.x ) a bf ( x ) = ( Inb ) bf ( * ) f' ( x ) ( 7 ) b ) 10 Ver - . . 5.1 5.2 Derivative of Exponential Function @2010 Julia & Teodoru Gugoiu - Page 2 of 3