Exam 1: Calculus & Modeling 1 (22M:021)
Prof. Keith Stroyan & T.A. Mike Winders	Tue 9 Oct 01	The Derivative

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1  Differentiate the following functions  y = f[x], a, b, c are constants

a) y = 5 (x^3) - 3 (x^5)

(dy/dx)=

b) y = (5/(x^3)) -  3 (x^(1/5))

(dy/dx)=

c) y = (x^5) Sin[x]

(dy/dx)=

c) y=(a (x^2) +  b x  +  c)(c (x^3)+ b (x^2)+a x)

(dy/dx)=

2  Differentiate the following functions  y = f[x], a, b, c are constants

a) y=Log[3 (x^2)+(6/x) -1]

(dy/dx)=

b) y = (e^(((x^2))))

(dy/dx)=

c) y = a Cos[b x]

(dy/dx)=

3  Tangent Lines

The explicit curve  y = f[x]  is shown below with  f[x] = ((x^3)/3) - x.  Calculate the x-values where the graph has a horizontal tangent line and sketch those tangents on the graph.

f[x_] = ((x^3)/3) - x; 
Plot[f[x], {x, -3, 3}, AxesLabel -> {"x", "y"}, 
        AspectRatio -> Automatic];
dy  =  f'[2] dx

4 Microscopic Views

a) Use the following Mathematica results to accurately sketch a microscopic view of the graph  y = f[x]  when your microscope is focused at the point  (1,1/e) ≈ (1.00, 0.368).

f[x_] = x (E^(-(x^3)));
f'[x]

f'[1]

f'[1.0]

b) Give an equation for the apparent curve you see in the microscope.  (You choose the variables, just make your choice clear.)

5 The Language of Change

In still air, a hemispherical mushroom cloud is spreading from the latest single cost-cutting explosion at the XYZ Nuclear Processing Plant.  The radius is increasing, but rate of increase of the radius is getting slower as the radius gets bigger.  Which of the following mathematical statements agree with this AND WHY OR WHY NOT?  (k is a positive constant that depends on the units)

(dr/dt)= -k (r^2)	
This "law of change" Agrees ( ) or Disagrees ( ) with the story above.  REASON:

(dr/dt)= k r
This "law of change" Agrees ( ) or Disagrees ( ) with the story above.  REASON:

(dr/dt) =  -(k/r)	
This "law of change" Agrees ( ) or Disagrees ( ) with the story above.  REASON:

(dr/dt) =  (k/(r^2))	
This "law of change" Agrees ( ) or Disagrees ( ) with the story above.  REASON: