Calculus Cheat Sheet.pdf

Combine rational expressions 2 1 1 1 1 lim lim 1 1 1

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+ + + = = Combine Rational Expressions ( ) ( ) ( ) ( ) 0 0 2 0 0 1 1 1 1 lim lim 1 1 1 lim lim h h h h x x h h x h x h x x h h h x x h x x h x + = + + = = = + + L’Hospital’s Rule If ( ) ( ) 0 lim 0 x a f x g x = or ( ) ( ) lim x a f x g x ± = ± then, ( ) ( ) ( ) ( ) lim lim x a x a f x f x g x g x = a is a number, or −∞ Polynomials at Infinity ( ) p x and ( ) q x are polynomials. To compute ( ) ( ) lim x p x q x ± factor largest power of x in ( ) q x out of both ( ) p x and ( ) q x then compute limit. ( ) ( ) 2 2 2 2 2 2 4 4 5 5 3 3 3 4 3 lim lim lim 5 2 2 2 2 x x x x x x x x x x x x →−∞ →−∞ →−∞ = = = Piecewise Function ( ) 2 lim x g x →− where ( ) 2 5 if 2 1 3 if 2 x x g x x x + < = ≥ − Compute two one sided limits, ( ) 2 2 2 lim lim 5 9 x x g x x →− →− = + = ( ) 2 2 lim lim 1 3 7 x x g x x + + →− →− = = One sided limits are different so ( ) 2 lim x g x →− doesn’t exist. If the two one sided limits had been equal then ( ) 2 lim x g x →− would have existed and had the same value. Some Continuous Functions Partial list of continuous functions and the values of x for which they are continuous. 1. Polynomials for all x . 2. Rational function, except for x ’s that give division by zero. 3. n x ( n odd) for all x . 4. n x ( n even) for all 0 x . 5. x e for all x . 6. ln x for 0 x > . 7. ( ) cos x and ( ) sin x for all x . 8. ( ) tan x and ( ) sec x provided 3 3 , , , , , 2 2 2 2 x π π π π L L 9. ( ) cot x and ( ) csc x provided , 2 , ,0, ,2 , x π π π π L L Intermediate Value Theorem Suppose that ( ) f x is continuous on [ a, b ] and let M be any number between ( ) f a and ( ) f b . Then there exists a number c such that a c b < < and ( ) f c M = .
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Calculus Cheat Sheet Visit for a complete set of Calculus notes. © 2005 Paul Dawkins Derivatives Definition and Notation If ( ) y f x = then the derivative is defined to be ( ) ( ) ( ) 0 lim h f x h f x f x h + = . If ( ) y f x = then all of the following are equivalent notations for the derivative. ( ) ( ) ( ) ( ) df dy d f x y f x Df x dx dx dx = = = = = If ( ) y f x = all of the following are equivalent notations for derivative evaluated at x a = . ( ) ( ) x a x a x a df dy f a y Df a dx dx = = = = = = = Interpretation of the Derivative If ( ) y f x = then, 1. ( ) m f a = is the slope of the tangent line to ( ) y f x = at x a = and the equation of the tangent line at x a = is given by ( ) ( )( ) y f a f a x a = + . 2. ( ) f a is the instantaneous rate of change of ( ) f x at x a = . 3. If ( ) f x is the position of an object at time x then ( ) f a is the velocity of the object at x a = . Basic Properties and Formulas If ( ) f x and ( ) g x are differentiable functions (the derivative exists), c and n are any real numbers, 1. ( ) ( ) c f c f x = 2. ( ) ( ) ( ) f g f x g x ± = ± 3.
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