MATH_152_Final_Study_Guide.pdf - MATH 15200/43 Final Study...

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MATH 15200/43: Final Study Guide The final will test everything we learnt throughout the quarter. Topics covered: Section 5.1-5.2 : Upper sum U f ( P ) and lower sum L f ( P ) of a function f under a partition P ; regular partition P n ; lim n →∞ U f ( P n ) and lim n →∞ L f ( P n ) as a way to determine if a function is integrable and compute b a f if the integral exists. Section 5.3 : First Fundamental Theorem of Calculus: F ( x ) = x a f ( t ) dt is an antiderivative of f if f is continuous. Compute d dx x a f ( t ) dt given an explicit function f . Section 5.4 : Second Fundamental Theorem of Calculus: b a f ( x ) dx = G ( b ) - G ( a ) where G is any antiderivative of f . Compute definite integrals b a f ( x ) dx . Section 5.5 : Compute the area of a region bounded by curve f and the x -axis between x = a and x = b ; Compute the area of a region bounded by two curves f and g . Section 5.6 : Indefinite integral f ( x ) dx ; compute indefinite integrals. Section 5.7 : Method of the u -substitution. Section 5.8 : Additional properties of integrals: general formula for d dx g 2 ( x ) g 1 ( x ) f ( t ) dt ; integral of even and odd functions; triangle inequality for integral; integral preserves inequality signs. Section 5.9 : Mean Value Theorem for Integral. Section 6.1 : Compute area of a region through summing up x-slices or y-slices. Section 6.2 : Compute volume of revolving solid using the Disk and Washer methods. Section 6.3 : Compute volume of revolving solid using the Disk and Shell methods. Section 7.1 : One-to-one functions, inverse functions, derivative of the inverse function. Section 7.2 : The natural logarithm function ln x , definition and properties. Section 7.3 : Derivative and Integral involving the natural log function ln x . Section 7.4 : Exponential function, its derivative and integral. Section 7.5 : Arbitrary powers, general log functions. Section 7.7 : The inverse trigonometric functions. Section 8.2 : Integration by parts. 1
Section 8.3 : Powers and products of trigonometric functions. Section 8.4 : Integrals with the term a 2 - x 2 , x 2 + a 2 , x 2 - a 2 . Section 9.1 : First-order linear differential equations. Section 9.2 : Separable differential equations. Trigonometric Identities (memorize all of them): 1. sin 2 x + cos 2 x = 1. 2. tan 2 x + 1 = sec 2 x . 3. cot 2 x + 1 = csc 2 x . 4. sin 2 x = 2 sin x cos x . 5. cos 2 x = cos 2 x - sin 2 x = 2 cos 2 x - 1 = 1 - 2 sin 2 x . Derivatives learnt in the class (memorize all of them): 1. ( x r ) = r · x r - 1 for any real number r . 2. (sin x ) = cos x . 3. (cos x ) = - sin x . 4. (tan x ) = sec 2 x . 5. (cot x ) = - csc 2 x . 6. (sec x ) = tan x sec x . 7. (csc x ) = - cot x csc x . 8. (arcsin x ) = 1 1 - x 2 . 9. (arccos x ) = - 1 1 - x 2 . 10. (arctan x ) = 1 1 + x 2 . 11. (arccot x ) = - 1 1 + x 2 . 12. (ln x ) = 1 /x . 13. ( e x ) = e x . 14. (log a x ) = 1 x · ln a for any constant a > 0. 2
15. ( a x ) = a x · ln a for any constant a > 0. Integrals learnt in the class (memorize all of them): 1. x r dx = 1 r + 1 · x r +1 + C for any constant real r = - 1. 2. 1 x dx = ln | x | + C . 3. e x dx = e x + C . 4. a x dx = a x ln a + C for any constant a > 0 and a = 1.
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