{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

practice math exam version 1

# practice math exam version 1 - Name Section Number TA Name...

This preview shows pages 1–4. Sign up to view the full content.

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

View Full Document
1. Find dy dx using implicit differentiation: (a) (3 points) x 4 y 5 – 5 xy = 100. Notation : We shall use the abbreviation y for dy dx . To proceed, we shall use the product rule. 4 x 3 y 5 + 5 x 4 y 4 y – 5( y + xy ) = 0 y (5 x 4 y 4 – 5 x ) = 5 y – 4 x 3 y 5 . Solving for y , we have: 35 44 54 55 yx y y x ′ = . (b) (3 points) cos( y ) = xy . -sin( y ) y = y + xy y (-sin( y ) – x ) = y . Solving for y , we have: sin( ) sin( ) yy y y xy x == −+ .
2. (4 points) A crate open at the top has vertical sides, a square bottom, and a volume of 108 cubic feet. If the crate has the least possible surface area, find its dimensions. What does this crate look like? We draw a picture: h b b We want to minimize surface area. The equation for surface area of a crate without a top is given by A = 4 bh + b 2 . (Otherwise, it would have been A = 4 bh + 2 b 2 .) We are given a constraint, namely V = hb 2 = 108. Using the constraint equation, we can solve for h in terms of b . Thus, h = 108/ b 2 . Substituting our expression for h into the formula for area, we get: 22 2 108 432 () 4 Ab b b b bb ⎛⎞ =+ = + ⎜⎟ ⎝⎠ . To find where this is minimized, we look at the derivative, set it equal to 0 and look for local minimum(s).

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 9

practice math exam version 1 - Name Section Number TA Name...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online