Solve all question and show work on a piece of paper neatly! Please

1. Find the limit lim x3e X-00 2. Differentiate 5/ 2 + sin(x) 3. If we start with A atoms of a radioactive material that has a "half-life" (the time it takes for half of the material to decay) of 500 years, then the number of radioactive atoms left after t years is r(t) = Ae-Kt where K = In2 500 . Calculate ro (t) and show that it is proportional to r(t) (that is, To (t) = br(t) for some constant b) 4. An expandable sphere is being filled with liquid at a constant rate from a tap (imagine a water balloon connected to a faucet). When the radius of the sphere is 3 inches, the radius is increasing at 2 inches per minute. How fast is the liquid coming out of the tap? 5. Calculate the area under the curve y = x on the interval [1, 3]. Consider only 6 subintervals with end points on the right. Use Riemann Sum Method and Show graph. 6. Integrate: ( 4 x 2 + 2x - 3 ) dx = cos 2 (x) . sin3 (x) dx 7. Evaluate: Li (ex - 2 ) dx x2 ( 2 In (x* ) + 73) dx 8. Find the area between the graph of the cosine and the horizontal axis x between 0 and It/2. 9. The velocity of a car after t seconds is 75 - 3t feet per second. a. How many seconds does it take for the car to come to a stop (velocity = 0)? b. How far does the car travel while coming to a stop c. How many seconds does it take the car to travel half the distance in part (b)? Show step by step solution. 10. How much work is done lifting a 20 pound bucket from the ground to the top of a 30 foot building with a cable which weighs 3 pounds per foot