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1. Finding function
Given \({dy \over dx}\), the function y=f(x) is found by integrating \(dy \over dx\) with respect to x and using the conditions given to calculate the constant of integration.
Step 1: integrate the derivative
Step 2: use the values given to calculate the constant of integration
Step 3: form the function
Example
Given that \(dy \over dx\) =2x-3 and that y=12 when x=2, find the equation of y in terms of x.
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\(y=\int {dy \over dx}dx \)
\( =\int (2x-3)dx \)
\( =x^2-3x+C \)
\( When \text { }x=2; \) \( 4-6+C=12 \)
\( C=52 \)
\( y=x^2-3x+{5 \over 2} \)
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Example
Given that \(dy \over dt\) =5t2-2 and that y=10 when t=-1, find the equation of y when t=1.
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\( y=\int {dy \over dt}dt \)
\( =\int (5t^2-2)dt \)
\( ={5 \over 3}t^3-2t+C \)
\( When \text { }t=-1; \) \( -{5 \over 3}+2+C=10 \)
\( C={29 \over 3 } \)
\( y={5 \over 3}t^3-2t+{29 \over 3} \)
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Example 3
Find the equation of a curve with gradient x2-2x and passing through point (1,-1).
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\( y=\int (x^2-2x)dx \)
\( =13x^3-x^2+C \)
\( At (1,-1); \) \( 13-1+C=-1 \)
\( C=-13 \)
\( y=13x^3-x^2-13 \)
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2. Calculating rate of change
Given the rate of change \(dy \over dt\), the function y=f(t) is found by integrating dydt with respect to t. As above, use the conditions given to calculate the value of the constant of integration.
Example
The rate of change of s with respect to t is given by dsdt=8t3 and when t=2, s=2. Find s in terms of t.
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\(s=\int {ds \over dt}dt \)
\( =8t^3dt \)
\( =2t^4+C \)
\( When t=2; \) \( 32+C=2 \)
\( C=-30 \)
\( s=2t^4-30 \)
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3. Other real life examples
Example
The change in the current of an electric circuit is given by
| \({dI \over dt}=-{100t \over t^2} \) |
where current, I, is measured in amperes and time t is measured in seconds.
Form the current function I(t) given that the current is 150 amperes at time 2 seconds.
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\( {dI \over dt}=-100t^{-2} \)
\( I=\int -100t^{-2}dt \)
\( =-100t-1-1+C \)
\( When t=2; \) \( 1002+C=150 \)
\( C=100 \)
\(∴I(t)=100t+100\)
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Example
A production manager finds that the cost to produce x number of items, in RM per item, is
| \({dC \over dx}=3.15+0.004x.\) |
Calculate the total cost of producing one hundred items if the fixed costs (that is the cost before production began) is RM450.
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\( Cost = \int (3.15+0.004x)dx \)
\( =3.15x+0.002x^2+C \)
\( When \text { }x=0; C=450 \)
\( Then, when \text { } x=100; \)
\( Cost =3.15(100)+0.002(100)2+450 \)
\( = RM 785 \)
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