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PHYS107

Problem Sheet 3: More Differentiation

This week we found out how functions like sin(x) and ln(x) were calculated by expressing them as a power series. For trigonometric functions this also leads to the “small angle

formulae” and approximations are sometimes helpful in doing larger calculations. Methods of differentiation are important, so let’s spend some time getting more practice. Don’t rush and remember to use these sheets and the support available in workshops to help you

fully understand. Make sure you get your work signed off in the session, before uploading a copy to Canvas by the end of the workshop day.

1)   Differentiate the following with respect to x:-

2)    Use implicit differentiation to differentiate the following, term by term with respect to x:

x2 + y2 + x sin(y) = 0

then rearrange to find

3)    Use logarithmic differentiation to find the derivative of:

4)    Use the Maclaurin Series overleaf to expand cos(x) as a power series.

Find the first three non-zero terms.

Show that the nth term (starting at n = 0) can be written as:

Use the ratio test, and find Does the series converge? For any x ?

5) And our guest question this week comes from the vectors part of the course…

A positive charge, +e, travelling with a velocity is passing through

a magnetic field given by

i)  Calculate the "Lorentz” force acting on the charge, given by

ii) Calculate the angle between each pair of vectors.