10 PROBLEMS ON CALCULUS

1. 
The table below shows the temperature (in °F) t hours after midnight in Phoenix on March 15. The table shows values of this function recorded every two hours. (10 points)
1.  Estimate the value of T'(10). Give units in your answer.
2.  What is the meaning of T'(10)?
t
0
2
4
6
8
10
12
14
T
73
73
70
68
73
80
86
89
2. 
Find the values of m and b that make the following function differentiable.
3. 
Find f'(x) for f(x) = cos4(5×2).
4. 
Find f'(x) for f(x) = ln(x2 + e3x).
5. 
Find  by implicit differentiation for ycos(x) = xcos(y).
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1. 
Find the x-coordinates of any relative extrema and inflection point(s) for the function f(x) = 6x(1/3) + 3x(4/3). You must justify your answer using an analysis of f ‘(x) and f “(x).
2. 
What is the maximum volume in cubic inches of an open box to be made from a 12-inch by 16-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides? Your work must include a statement of the function and its derivative. Give one decimal place in your final answer.
3. 
The position function of a particle in rectilinear motion is given by s(t) = 2t3 – 21t2 + 60t + 3 for t ≥ 0 with t measured in seconds and s(t) measured in feet. Find the position and acceleration of the particle at the two instants when the particle reverses direction. Include units in your answer.
4. 
Water is drained out of tank, shaped as an inverted right circular cone that has a radius of 6cm and a height of 12cm, at the rate of 3 cm3/min. At what rate is the depth of the water changing at the instant when the water in the tank is 9 cm deep? Give an exact answer showing all work and include units in your answer.
5. 
The side of a square is measured to be 16 ft with a possible error of ±0.1 ft. Use linear approximation or differentials to estimate the error in the calculated area. Include units in your answer.

 

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