M2Solns

# M2Solns - MATH 31A (Butler) Midterm II, 19 February 2010 1....

This preview shows pages 1–3. Sign up to view the full content.

MATH 31A (Butler) Midterm II, 19 February 2010 1. Find the unique value c that satisﬁes the Mean Value Theorem for the function f ( x ) = arctan(sin x ) for x between a = 0 and b = π/ 2. (Your answer will involve an arcsin or arccos term. Hint: sin 2 x + cos 2 x = 1; the quadratic formula is awesome!) First we recall the statement of the Mean Value Theorem, namely if our function is continuous and diﬀerentiable between points a and b then there is some c between a and b so that f 0 ( c ) = f ( b ) - f ( a ) b - a In our case we need to have f 0 ( c ) = f ( π 2 ) - f (0) π 2 - 0 = arctan(sin π 2 ) - arctan(sin0) π 2 = arctan(1) - arctan(0) π 2 = π 4 - 0 π 2 = 1 2 . Taking the derivative of the function (using the chain rule and the rules for the derivatives of the sine and arctangent functions) we have f 0 ( x ) = 1 1 + (sin x ) 2 · cos x = cos x 1 + sin 2 x . So we need f 0 ( c ) = cos c 1 + sin 2 c = 1 2 or 2cos c = 1 + sin 2 c. Using sin 2 c + cos 2 c = 1 this can be rewritten as 2 cos c = 2 - cos 2 c or cos 2 c + 2 cos c - 2 = 0 . This is a quadratic (but with cos c instead of x ) so we can use the quadratic formula to ﬁnd cos c = - 2 ± q 2 2 - 4 · 1 · ( - 2) 2 = - 2 ± 12 2 = - 1 ± 3 . But - 1 - 3 < - 1 and so can never be the cosine of an angle. So we must have that cos c = - 1 + 3 or c = arccos( - 1 + 3) .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2. You have recently been hired as the chief architect for one of the pyramids con- structed by Pharaoh Sneferu in ancient Egypt. After some consultation the pharaoh has agreed to a pyramid design that is 500 cubits wide and 300 cubits high (a cubit is the system of measurement used in ancient Egypt). The volume of a pyramid is 1 3 b 2 h where b is the length of one side of the base and h is the height; so that the pyramid will require 25,000,000 cubic cubits of stone. After getting in touch with your stone contractor you discover that there are only 23,000,000 cubic cubits of stone available. The pharaoh gives the go ahead to build a (slightly) smaller pyramid, but with the same proportions as before. Estimate how many cubits smaller the base of the pyramid will end up being. We have that the volume of the pyramid is
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 04/23/2010 for the course MATH 262-181-20 taught by Professor Butler during the Winter '10 term at UCLA.

### Page1 / 5

M2Solns - MATH 31A (Butler) Midterm II, 19 February 2010 1....

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online