Exam 2: Calculus & Modeling 1 (22M:021)   2 Nov 01	
Mike Winders & Keith Stroyan	Applications of The Derivative

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(({{1  }, {     }}),({{2  }, {    }}),({{3  }, {    }}),({{4  }, {    }}),({{5  }, {    }}),({{Tot.}, { }}))

1 Knee Jerk

A function of time,  y =  y[t]  satisfies:

y[0]  =  5
 &				
	(dy/dt) = -7 y

Give the exact value of  y[4] = ? and explain your formula.  (Note:  Log[2]  is exact, 0.693147, is approxiamte.  Sin[1]  is exact, 0.841471,  is approximate.)

2 Max-min

Find the max and min of the function  f[x] = (x^3) - 6 x + 5  for  x  in the interval  [-2,3].  Does  f[x]  have both a max and min?  Explain your reasoning carefully.

f[x_] = (x^3) - 6 x + 5;
f'[x]
Solve[f'[x] == 0]
Plot[f[x], {x, -2, 3}]

f[-1]
f[-2]
f[3]
f[1]

3 Max-min

What are the largest and smallest values that the function f[x] = ((x^2)/(3 + (x^2)))could attain for all real values of  x ?

Clear[f, x];
f[x_] = (x^2)/(3 + x^2);
Together[f'[x]]

Solve[f'[x] == 0]

f'[-1]

f'[1]

The slope table is down, over, up, so x=0 is a minimum, but there is no maximum ATTAINED.

Limit[f[x], x -> ∞]

The supremum of the function is  1, but it is not attained.

Plot[f[x], {x, -10, 10}, PlotRange -> {0, 1}, 
        PlotStyle -> RGBColor[1, 0, 0]];

Does  f[x]  actually attain these values or are they limiting cases?  Prove your assertions for example by giving a sketch of the graph for  x  in [0,∞) with sufficient supporting facts.  You are given the derivative  f'[
          x]  =  ((6 x)/(((3 + (x^2)))^(2 )))

4 Max-min

If they exist, find the dimensions of rectangles of largest and smallest area that can be inscribed in a semicircle of radius  25  horizontally as shown. Explain your results.

Clear[x, y, s];
x =  25 Cos[s] 
y = 25 Sin[s] 
circle = 
        ParametricPlot[{x, y}, {s, 0, π}, Ticks -> None, 
          PlotStyle -> RGBColor[1, 0, 0]];
s = π/8;
Show[{circle, 
          Graphics[{RGBColor[0,  .4, 0], 
              Polygon[{{x, 0}, {x, y}, {-x, y}, {-x, 0}}]}], 
          Graphics[{Line[{{0, 0}, {x, y}}], 
              Text["25", {x, y}/2 + {0,  .25}]}]}];

5 Four functions and their derivatives are:

(Subscript[f, 1])[
        x] = x (e^x)  and  (Subscript[f, 1])'[
            x] = (1 + x) (e^x)  and   (Subscript[f, 1])''[
              x] = (2 + x) (e^x)

(Subscript[f, 2])[
        x] = (x^2) (e^x)  and  (Subscript[f, 2])'[
            x] = x (x + 
                2) (e^x)   and  (Subscript[f, 2])''[
              x] = (2 + 4 x + (x^2)) (e^x)

\(TextForm\`f\_3[
        x] = \(x\ e\^\(x\^2\)\ \ and\ \ \(f\_3'\)[
            x] = \(\((1 + 
                2\ x\^2)\)\ e\^\(x\^2\)\ \ \ and\ \ \(\(f\_3'\)'\)[x] = 
          2\ x\ \((3 + 2\ x\^2)\)\ e\^\(x\^2\)\)\)\)

\(TextForm\`f\_4[
        x] = \(e\^\(x\^3\)\ \ and\ \ \(f\_4'\)[
            x] = \(3\ \(x\^2\) e\^\(x\^3\)\ \ \ and\ \ \(\(f\_4'\)'\)[x] = 
          3\ x\ \((2 + 3\ x\^3)\) e\^\(x\^3\)\)\)\)

(NOTE:  Recall that (e^x) is always positive, (e^(-x)) = 1/(e^x).)

f1[x_] = x (E^x)
f2[x_] = (x^2) (E^x)
f3[x_] = x (E^((x^2)))
f4[x_] = (E^((x^3)))

Factor[f1'[x]]
Factor[f2'[x]]
Factor[f3'[x]]
Factor[f4'[x]]

Factor[f1''[x]]
Factor[f2''[x]]
Factor[f3''[x]]
Factor[f4''[x]]

\(
\(grf1 = 
        Plot[f1[x], {x, \(-2\), 3/2}, PlotRange -> {\(-2\), 5}, 
          AxesOrigin -> {0, 0}, AxesLabel -> {"x", "y"}, 
          Ticks -> None, PlotLabel -> "Graph d        "];\)\)
grf2 = 
        Plot[f2[x], {x, -2, 3/2}, PlotRange -> {-2, 5}, 
          AxesOrigin -> {0, 0}, AxesLabel -> {"x", "y"}, 
          Ticks -> None, PlotLabel -> "Graph b        "];
grf3 = 
        Plot[f3[x], {x, -2, 3/2}, PlotRange -> {-2, 5}, 
          AxesOrigin -> {0, 0}, AxesLabel -> {"x", "y"}, 
          Ticks -> None, PlotLabel -> "Graph c        "];
grf4 = 
        Plot[f4[x], {x, -2, 3/2}, PlotRange -> {-2, 5}, 
          AxesOrigin -> {0, 0}, AxesLabel -> {"x", "y"}, 
          Ticks -> None, PlotLabel -> "Graph a        "];
Show[GraphicsArray[{{grf4, grf2}, {grf3, grf1}}]];

a)  Which functions and which graphs have a positive derivative at  x = 0?  
Function \(\(f\_?\)[x]\)    and Graph ?

b)  Which functions and which graphs have a positive derivative for all  x < 0?   
Function \(\(f\_?\)[x]\)    and Graph ?

c)  Which functions and which graphs have a positive SECOND derivative at  x = 0?  
Function \(\(f\_?\)[x]\)    and Graph ?

d)  Which functions and which graphs have a negative second derivative for x<0, x≈0, and positive second derivative for x>0, x≈0,?  Function \(\(f\_?\)[x]\)    and Graph ?

e)  Write "(Subscript[f, 1])[x]"  on its graph, "(Subscript[f, 2])[x]'" on its graph,"(Subscript[f, 3])[x]"  on its graph, and "(Subscript[f, 4])[x]"  on its graph.