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finalreviewf10

# finalreviewf10 - t 1 2 t 2 ml/sec where t is the time...

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MAC 2311, Fall 2010: Some Review Problems (Lectures 34 and 35) Final covers lectures 1 - 35 1. Evaluate each integral: a) Z ( p x + 1) 2 x dx b) Z 3 x + 3 p x 2 + 2 x + 5 dx c) Z e 4 1 ln x 2 x dx d) Z 3 csc(2 x ) cot(2 x ) dx e) Z e 1 2 x ± 1 (2 x ± 1) 2 dx f) Z e 2 x e x ± 1 dx g) Z cos x 1 + sin 2 x dx h) Z 0 ± 4 x p x + 4 dx i) Z cot xdx 2. The slope of the tangent line to the curve y = f ( x ) at any point is given by 2 sin ± x 2 ² cos 2 ± x 2 ² . If f (0) = 1, ±nd the function f ( x ). 3. Find the area under the graph of f ( x ) = ( x 2 x < 0 e x 3 x ² 0 on [ ± 2 ; 3 ln 4]. 4. Evaluate Z e 1 f ( x ) dx if a) f ( x ) = 2 x (3 ln x ± 2) and b) f ( x ) = 2 x (3 ln x ± 2) 2 . 5. Find f ( ± 1) if f 00 ( x ) = 2 ( x + 4) 2 , f 0 ( ± 2) = 3 and f ( ± 3) = 2. 6. Evaluate d dx Z p x x 3 t 2 sin t 2 dt . 7. Suppose that f is an even function and g is odd, both are continuous, and g ( x ) ² 0 on [0 ; 10]. If Z 10 5 g ( x ) dx = 10, Z 5 0 g ( x ) dx = 5, Z 5 ± 5 f ( x ) dx = 12 and Z 10 0 f ( x ) dx = 20, ±nd: a) Z 10 ± 5 [ f ( x ) + g ( x )] dx b) the area of region bounded by the x -axis and g ( x ) on [ ± 5 ; 10] 8. Consider the area given by Z 2 0 x x 2 + 2 dx . a) Approximate the area using a Riemann sum with n = 4 and using the midpoint of the interval [ x i ± 1 ;x i ] as x ² i . b) Find the exact value of the integral. 9. A soft drink dispenser pours a soft drink at the rate of f ( t ) = 20
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Unformatted text preview: t 1 + 2 t 2 ml/sec, where t is the time elapsed in seconds. How much of the drink is dispensed in the ±rst 2 seconds? 10. A particle moves along a straight line with velocity v ( t ) = t 2 ± 2 t ± 8. Find the displacement of the particle on the time interval [0 ; 5], and the total distance traveled by the particle. 11. A population is changing at the rate dP dt = 50 t ± 100 t 3 = 2 where P is the population in thousands after t years. Find the net change in population in the ±rst 4 years ( t = 0 to t = 4). 12. The price of a particular model of a Toyota is increasing at a rate of 3 t p 3 t 2 + 4 thousand dollars t years after its introduction. If the base price of the car was \$18,000 when it was ±rst introduced, what will the base price be of the same model two years later?...
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