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definition of derivative

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1. Find the following limits.

a) Limit as x approaches 3 of: (x^2-x-6)/(x-3)
b) Limit as theta approaches 0 of: (sin(3 theta))/(sin(5 theta))

2. Using definition of derivative find the derivative of:

a) f(x)= 4x^2-6x-5
b) f(x)= (1-x)/(x+1)

3. Differentiate and simplify:

a) y=cos^3(2x^2+3)
b) y=ln sqrt(1-x^2)
c) y=1/((x^3+1)^4)
d) y=x^2 sqrt(1-x^2)

4. Find the equation of the tangent line in slope intercept form of the curve given by:

4x^2y+y^3-2x+14 passing through (1,2)

5. Let s(t)= 3 sqrt(t^2+5) be the displacement function of a particle with distance in feet and time in second. At the end of the 2nd second,

a) what is the particle's velocity?
b) what is its acceleration?

6. At a certain instant, a small balloon is released from the ground level at a point 75 feet away from an observer also at ground level. If the balloon goes straight up at a rate of 2.5 feet per minute, how rapidly will it be moving away from the observer 40 seconds later?

7. Sand is being poured on a conical pile at the rate of 12 cubic feet per minute, the height of the pile being three times its diameter throughout. At what rate is the height increasing when the pile is 8 feet high?

8. Find limit as x approaches 2 where

f(x)=1+2x, if x is less or equal to 2 and x^2+2 if x is greater than 2