Applications of Derivatives and Rate of Change

1.) (d/dx)(xe^(lnx^2))=?

2.) If x=e^(2t) and y=sin(2t), then (dy/dx)=?

3.) If y=xy+x^2+1, then when x=-1, (dy/dx) is ?

4.) A particle moves along the x-axis so that its acceleration at any time is a(t)=2t-7. If the initial velocity of the particle is 6, at what time t during the interval 0≤t≤4 is the particle farthest to the right?

5.) The position of an object attached to a spring is given by the formula: y(t)=(1/6)cos(5t)-(1/4)sin(5t), where t is time in seconds. In the first 4 seconds, how many times is the velocity of the object equal to 0?

6.) If (dy/dx)=(1+lnx)y and if y=1 when x=1, then y= ?

7.) Let f be a twice differential function such that f(1)=2 and f(3)=7. Which of the following must be true for the function f on the interval 1≤x≤3?
I. The average rate of change of f is 5/2
II. The average value of f is 9/2
III. The average value of f' is 5/2

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