# Assorted Differentiation and Tangent to Curve Problems

A. i) Differentiate the equations given as items 21 and 22 on your worksheet.

ii) Refer to the formula given as item 23 of your worksheet.

The equation relates to one particular machine in an engineering workshop. The machine sots C pounds to lease each week according to the formula and 't' is the number of hours per week worked by the machine. The rate of increase of cost during the week is given by dC/dt.

a) Find a general expression for dC/dt.

b) Use your expression to calculate after how many hours (t) the rate of increase of cost exceeds 1.50 pounds.

iii) Integrate the equations given as items 24 and 25 on your worksheet.

B. Refer to the worksheet item 26 wich gives an equation which relates distance (s) in metres and time (t) in seconds for a moving body.

i) Using the equation given as item 26, find an equation for the acceleration, dv/dt of the body.

ii) Use your equation from part B i) to find the acceleration of the body in m/s^2 after the time shown as item 27 on the worksheet.

iii) On your worksheet, refer to the equation given as item 28. Given that:

Power = 2.3 the integral of V dI Watts, calculate the power (P) when the current (I) changes within the range given as item 29 on your worksheet.

C. i) Refer to the equation given as item 30 on your worksheet. Plot the graph of the equation over one cycle using co-ordinate points at intervals of 30 degrees.

ii) Draw a tangent to the curve at theta = 120 degrees. Estimate the value of the gradient of the tangent at this point.

iii) Confirm your answer above by using the rules of differentiation.

iv) Explain why your answers to parts ii) and iii) above are not identical and discuss whether in your opinion they are within acceptable limits.

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#### Solution Preview

Note: In your previous posting in any places, it was not very clear what you meant; I gave you different answers based on what I can infer from your questions.

Now that you posted clear questions, I see that some of the questions were misinterpreted by me. Given below are the correct answers.

I will use the "^" for exponents; dy/dx for derivative

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Question 4:

A)

i)

21. y = 8/(5x^2) = (8/5)* x^(-2)

dy/dx = (8/5) * (-2) * x^(-3) = (-16/5)X^-3

22. y = 4cosx-3e^x

dy/dx = -4 Sinx - 3e^x

ii)

23. a) Find a general expression for dC/dt.

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#### Solution Summary

Assorted differentiation and tangent to curve problems are solved. The solution is concise and helpful.