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Given that dy/dx = 6x^5 - 3x + 4, and y = 14 when x = 2, express y in terms of x. - Scottish Highers Maths - Question 10 - 2018

Question 10

Given that dy/dx = 6x^5 - 3x + 4, and y = 14 when x = 2, express y in terms of x.

Worked Solution & Example Answer:Given that dy/dx = 6x^5 - 3x + 4, and y = 14 when x = 2, express y in terms of x. - Scottish Highers Maths - Question 10 - 2018

Step 1

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Answer

To express y in terms of x, we first need to integrate the derivative:

$y = \int (6x^5 - 3x + 4) \, dx$

This gives:

$y = \frac{6}{6}x^6 - \frac{3}{2}x^2 + 4x + C$

Thus,

$y = x^6 - \frac{3}{2}x^2 + 4x + C$ .

Step 2

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Answer

Using the condition y = 14 when x = 2:

$14 = (2)^6 - \frac{3}{2}(2)^2 + 4(2) + C$

Calculating the right-hand side:

$14 = 64 - \frac{3}{2}(4) + 8 + C$

Simplifying:

$14 = 64 - 6 + 8 + C$ $14 = 66 + C\implies C = 14 - 66 = -52.$

Therefore, the constant is:

$C = -52.$

Step 3

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Answer

Substituting C back into our equation:

$y = x^6 - \frac{3}{2}x^2 + 4x - 52.$

Thus, we have expressed y in terms of x.

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