Topic 4 - Term Test 2

Question 1

Differentiate

lim as x approaches 0+
x^(3sin(x))

Answer 1

take ln of BOTH sides

ln(L)=limx-0+ 3sin(x)ln(x)

Turn into fraction by taking reciprocal
ln(x)
——
1/3sin(x)

take derivative of both and simplify and evaluate the limit

don’t forget that youre answer is in the form ln(L) = 0

Question 1

A lighthouse is located on a small island, 3 km away from the nearest point P on a straight shoreline. Its light makes
four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P

Answer 1

You’re going to use trig and the fact that revolutions indicates an angle.

Draw the shore and the lighthouse parallel to each other with the distance of 3km.

You should know that dtheta/dt = 8pi (for 2pi rad per rev and 4 revs) and you want dx = dt when x = 1.

So you are going to use tantheta = x/3 and then differentiate and solve. Easier to solve for the sectheta than just theta becuase you can find the hypotenuse using pythagorean theorem