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Question: Functions & Chain Rule

Question:Functions & Chain Rule

paulmcquad 60 Maple
procedure homework chain-rule
May 10 2020
0

Hi,

I was hoping someone can tell me if there's a better way for doing this math.

I think i was doing this wrong so i posted the full results. The answers r correct but i might be doing this the long way.

Thanks in Advance.
 

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Section 1.6: Combinations of Functions
> 

 

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Example 1: - Sum Rule
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"ex1f(x):= 2 x+1"

proc (x) options operator, arrow, function_assign; 2*x+1 end proc

 (1)
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"ex1g(x):=x^(2)+2 x-1"

proc (x) options operator, arrow, function_assign; x^2+2*x-1 end proc

 (2)
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ex1f(x)+ex1g(x)

x^2+4*x

 (3)
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eval(x^2+4*x, [x = 2])

12

 (4)
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Example 2: - Difference Rule
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"ex2f(x):=2 x+1"

proc (x) options operator, arrow, function_assign; 2*x+1 end proc

 (5)
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"ex2g(x):=x^(2)+2 x-1"

proc (x) options operator, arrow, function_assign; x^2+2*x-1 end proc

 (6)
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ex2f(x)-ex2g(x)

-x^2+2

 (7)
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eval(-x^2+2, [x = 2])

-2

 (8)
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Example 3: - Product Rule
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"ex3f(x):=x^(2)"

proc (x) options operator, arrow, function_assign; x^2 end proc

 (9)
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"ex3g(x):= x-3"

proc (x) options operator, arrow, function_assign; x-3 end proc

 (10)
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ex3f(x)*ex3g(x)

x^2*(x-3)

 (11)
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collect(x^2*(x-3), x)

x^3-3*x^2

 (12)
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eval(x^3-3*x^2, [x = 4])

16

 (13)
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Example 4: - Quotient Rule
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"ex4f(x):=sqrt(x)"

proc (x) options operator, arrow, function_assign; sqrt(x) end proc

 (14)
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"ex4g(x):=sqrt(4-x^(2))"

proc (x) options operator, arrow, function_assign; sqrt(4-x^2) end proc

 (15)
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ex4f(x)/ex4g(x)

x^(1/2)/(-x^2+4)^(1/2)

 (16)
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Example 5: - Composition - Chain Rule

 

This is an example of the chain rule.

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"ex5t1(x):=sqrt(x):"

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"ex5t2(x):=x-1:"

Function in another function:

 

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ex5t1(ex5t2(x))

(x-1)^(1/2)

(1.1)
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eval((x-1)^(1/2), [x = 2])

1

(1.2)
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plot((x-1)^(1/2))

 

Example 6: - Composition - Chain Rule

 
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Insert*two*functions

 

"ex6f(x):=x+2:"

"ex6g(x):=4-x^(2):"

 

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ex6f(ex6g(x))

-x^2+6

(2.1)
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eval(-x^2+6, [x = 0])

6

(2.2)
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eval(-x^2+6, [x = 1])

5

(2.3)
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``

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ex6g(ex6f(x))

4-(x+2)^2

(2.4)
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eval(4-(x+2)^2, [x = 0])

0

(2.5)
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eval(4-(x+2)^2, [x = 1])

-5

(2.6)
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``

Example 7: Find Domain with Chain Rule

 
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Insert two Functions:

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"ex7f(x):=x^(2)-9:"

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"ex7g(x):=sqrt(9-x^(2)):"

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``

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ex7f(ex7g(x))

-x^2

(3.1)
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plot(-x^2)

 

[-3, 3]

[-3, 3]

(3.2)
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``


 

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