Calculus: The Language of Change

Exercise Set 6.2
The Superposition Rule

by Michelle Nguyen

Exercise 1) Basic Superposition Drill

1) Find  (dy/dx) for each of the following functions y = y[x].  (The letters a, b, c, and h denote constants or parameters, and e is the natural  base for logs and exponentials.)

a)

y= 7(x^4)

Solution  of part a

By observing the equation  in part  (a),   we know  the variable in this equation is x, the constant coefficient  is 7, and  the  exponent is 4.  Using the  step-by- step Differentiation  with WalkD[.,.].

\(\(\(WalkD[7  x\^4, x]; \)

\)\)

b)

y= -5 (x^2)

Solution of part b

By observing the equation  in part  (b),  we know  the variable in this equation is x, the constant  coefficient is -5, and  the  exponent is 2.  Using the  step-by- step Differentiation  with WalkD[.,.].

\(\(\(WalkD[\(-5\) x\^2, x]; \)

\)\)

c)

y= 7x + (x^4)

Solution of part c

\(\(\(WalkD[7  x\  + x\^4, x]; \)

\)\)

d)

y= 8 (x^3) - 5 (x^2) + 4 x + 1

Solution of part d

\(\(\(WalkD[8\ x\^3 - 5\ x\^2 + 4\ x + 1, x]; \)

\)\)

e)

y= (7/(x^4))

Solution of part e

\(\(\(WalkD[7\/x\^4, x]; \)

\)\)

f)

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

Solution of part f

\(\(\(WalkD[\(-5\) \@x\%3, x]; \)

\)\)

g)

y= 7(x^(1/5))- 3(1/(x^4))

Solution of part g

WalkD[7 ( x^(1/5))- 3(1/(x^4)),x];

h)

y= 7\(TraditionalForm\`\@x\/\@x\%3\) - 5(1/(Sqrt[x]))

Solution of part h

WalkD[7\(\@x\/\@x\%3\) - 5(1/(Sqrt[x])),x];

i)

y= 7 (e^x) - 3 Log[x]

Solution of part i

WalkD[7 (E^x)-3 Log[x],x];

j)

y= (7/(x^e)) + 3 Log[(1/x)]

Solution of part j

WalkD[(7/(x^E))+3 Log[(1/x)],x];

k)

y= e ( x^(1/5)) - (π/(x^4))

Solution of part k

WalkD[ E ( x^(1/5)) - (π/(x^4)),x];

l)

y=3 Sin[x]-Cos[7 e]

Solution of part l

WalkD[ 3 Sin[x]-Cos[7 E],x];

m)

y=f[x] = a + bx + (cx^2) + (hx^3)

\(\(\(WalkD[x + x\^2 + x\^3, \ x]; \)

\)\)

n)

y= f[x]= 7 Cos[x] - 3 Sin [x]

Solution  of part n

\(\(\(WalkD[7\ Cos[x]\  - \ 3\ Sin\ [x], x]; \)

\)\)

Exercise 2)

Let f[x] = (Sqrt[x ]), let g[x]= (x^2), let a = 5, and let b = π.  Write out each  step of the proof of the  Superposition Rule that  appears  above  except  write the  steps with these specific functions and constants.

Solution 

Refer to page 83 for the  proof of The Superposition Rule!  Then follow the  rules and insert the given f[x], g[x], a, and b,  whenever you see these symbols appear in the proof.  In other words, just plug these values into  the original equations.
 
The  proof has four main equations:

1) h[x+δx] - h[x] = ( a f[x+δx] + bg[x+δx] - (af[x] + bg[x])
                             = ( 5(Sqrt[[x + δx]])  +π[x + δx]^2 - (5(Sqrt[x]) + π(x^2))
                             
2)                          = (f [x+δx] - f[x] + b(g[x+δx] - g[x])
                             = 5((Sqrt[[x + δx]])- (Sqrt[x]) )+ π((([x+δx])^2) - (x^2))

3)                          = a ( f'[x] δx+ (Subscript[ϵ, 1])δx) + b(g'[x]δx + (Subscript[ϵ, 2])δx)
                             = 5( (1/(2 (Sqrt[x])))δx + (Subscript[ϵ, 1])δx) + π( 2xδx + (Subscript[ϵ, 2])δx)

4)                          = (af'[x] +b g'[x])δx + (a(Subscript[ϵ, 1])+ b(Subscript[ϵ, 2]))δx
                             = (5 (1/(2 (Sqrt[x])))+ π2x)δx + (5(Subscript[ϵ, 1 ])+ π(Subscript[ϵ, 2]))δx
We know that (5(Subscript[ϵ, 1 ])+ π(Subscript[ϵ, 2])) ≈ 0 so we have
		((d (5 (Sqrt[x]) + π (x^2)))/dx) = 5 (1/(2 (Sqrt[x])))+ π2x

Exercise 3) Other Variables

a)

v = (u^2) + 2 u + 2

Solution  of part a

WalkD[(u^2) + 2 u + 2, u];

b)

u = 4 (Sqrt[v]) - (π/(v^2))

Solution  of part b

WalkD[ 4 (Sqrt[ v]) - (π/(v^2)), v];

c)

y = 3 u (Sqrt[u]) + (5/(u^2))

Solution  of part b

WalkD[3 u (Sqrt[u]) + (5/(u^2)) , u];

d)

u = (v^(1/4)) - (1/(v^(1/3)))

Solution  of part d

\(\(\(WalkD[v\^\(1/4\) - 1\/v\^\(1/3\), v]; \)
\)\)

e)

y = Cos[θ] - Sin [θ]

Solution  of part d

\(\(\(WalkD[Cos[θ]\  - \ Sin\ [θ], θ]; \)

\)\)

f)

u =Log[v] - (e^v)

Solution  of part f

\(\(\(WalkD[Log[v] - E\^v, v]; \)
\)\)