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Math 151/ Sample Final Exam Part I [100 pts. [5 pts.] Find the linearization of f (x) = x + 1 + sin x at x = 0 . [35 pts.] Evaluate each of the...

find the linearization of f(x) sqart x+1 +sinx at x=0

Math 151/ Sample Final Exam Part I [100 pts.] 1. [5 pts.] Find the linearization of f ( x ) = x + 1 + sin x at x = 0 . 2. [35 pts.] Evaluate each of the following limits. (a) lim h 0 sin( x + h ) - sin x h (b) lim x →∞ ln(1 + 2 e x ) x (c) lim x →∞ x (1 /x ) (d) lim x 0 (1 + 3 x ) 2 /x (e) lim x 0 + ± 3 x + 1 x - 1 sin x ² 3. [6 pts.] Solve the initial value problem: y 0 = 9 x 2 - 4 x + 5 , y ( - 1) = 0 . 4. Consider the function f ( x ) = 2 x 2 over the interval [0 , 3] . (a) [4 pts.] Find a formula for the Riemann sum of f over the given interval by dividing the interval into n equal subintervals and using the right–endpoints of the subintervals. (Your formula must be an expression of n only.) (b) [2 pts.] Take the limit as n → ∞ of the Riemann sum you found in part (a). 5. [6 pts.] Find the average value of the function f ( x ) = ( x - 1) 3 over the interval [ - 2 , 0] . 6. [42 pts.] Evaluate each of the following definite integrals. (a) Z 0 - 3 p 9 - x 2 dx (b) Z π/ 2 0 5 cos x - 2 sin x dx (c) Z ln 5 ln 2 3 e x dx (d) Z 1 1 / 3 4 1 + x 2 dx (e) Z 8 3 4 x x 2 + 1 dx (f) Z 4 1 (1 + x ) 3 x dx ————————————- Good Luck! ————————
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