Review for Exam 1 on the Derivative: (Exam Tue 9 Oct)
 Calculus & Modeling 1 Prof.: Keith Stroyan & T.A.: Mike Winders

Also see the Differentiation Quizzes with solutions in the Course Files of your UNIX account in B5.

1  Differentiate the following functions  y = f[x]

a) y  =  (x^(1/3)) 

(dy/dx)=

b) y  =  (((x^2))^(1/3))

(dy/dx)=

c) y  =  (x^(1/5)) 

(dy/dx)=

d) y  =  (1/(x^2)) 

(dy/dx)=

e) y = (1/((x^3))) 

(dy/dx)=

f) y = (1/(Sqrt[x])) 

(dy/dx)=

g) y = (x^3) Cos[x]

(dy/dx)=

h) y = (x^3) Log[x]

(dy/dx)=

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

(dy/dx)=

j) y = Log[3 (x^5) + 5 (x^3) + 15]

(dy/dx)=

k) y = Log[2 (x^3) + 3 (x^2) + 6]

(dy/dx)=

l) y=Cos[3 (x^2)+(6/x) -1]

(dy/dx)=

m) y = (e^Cos)[x]

(dy/dx)=

n) y = (e^Sin)[x]

(dy/dx)=

o) y=a (x^2)+ Sin[b x],  (a & b constants)

(dy/dx)=

o) v=a (e^(2  u))+ Log[b u],  (a & b constants)

(dv/du)=

2  Tangent Lines

The explicit curve  y = f[x]  is shown below with  f[x] = ((x^3)/3) - x.  Find the equation of the line tangent to the curve at the point where  x=2  and sketch that line on the graph below.  You may use coordinates of your choice, but specify that choice clearly and if they are not (x,y)-coordinates use them in your plot.

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

3

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).

N[1/E]

f[x_] = x (E^(-(x^2)));
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.)

4 Microscopes & f[x+δx]-f[x]=f'[x]·δx + ε·δx

The graph  y = f[x]  at  (dx,dy) = (x + (4/3), y - (337/324)) with  δx = ((1/4)) is shown below on the left and magnified by  4  on the right side.  The formula  f[x+δx]-f[x]=f'[x]·δx + ε·δx  at this specific  x  and with this  δx  tells us something about these graphs.

a)  Where is the particular  ε  shown to scale on the figure above? (Mark it clearly on the figure.)

Mathematica can find the general formula for  ε  in the microscope equation as follows:

f[x_] = (((x^4) + 1)/4);
ε=Factor[Expand[((f[x + δx]-f[x]-f'[x]δx)/(δx))]]

b)  What is the exact numerical value for this figure of  ε =

(plug in things like (4/3)and (1/4), but no need to evaluateyour expression approximately on a calculator)

c)  What is the limit of  ε  as  δx → 0 in general and for this specific point?  Why?

(Underscript[Lim, δx -> 0]) ε  =  
        (Underscript[Lim, δx -> 0])  ...

d)  What does this limit of  ε  mean about figures like the one above?  Does your answer depend on the particular  x-center of the microscope?

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.  As the radius of the cloud grows, the volume of the cloud grows even faster, so the 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	
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: