psCalculus2Ans

# psCalculus2Ans - Fall 2007 Problem Set #10- Answer key...

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

Fall 2007 ARE211 Problem Set #10- Answer key Second Calculus Problem Set (1) Consider the function f ( x, y, z ) = xyz , with y = x 2 and z = x 1 / 3 . (a) Rewrite f as a function g : R R alone and compute g p ( · ). Using g p , approximate the change in f when x increases by 0.1 units, starting from (8 , 64 , 2). Ans: g ( x ) = f ( x, x 2 , x 1 / 3 ) = x × x 2 × x 1 / 3 = x 10 / 3 so that g p ( x ) = 10 / 3 x 7 / 3 g p (8) = 10 / 3 × 8 7 / 3 = 10 / 3 * 2 7 = 426 . 6667 dg = df = 0 . 1 × 10 / 3 × 128 = 42 . 6 . (b) Compute the total derivative of f with respect to x . Using the total derivative, ap- proximate the change in f when x increases by 0.1 units, starting from (8 , 64 , 2). Ans: d f d x = f x + f y d y d x + f z d z d x = yz + xz × 2 x + yx × 1 / 3 x - 2 / 3 = x 7 / 3 + x 4 / 3 × 2 x + x 3 × 1 / 3 x - 2 / 3 = x 7 / 3 (1 + 2 + 1 / 3) = 10 / 3 x 7 / 3 so that df = d f (8) d x = 0 . 1 × 10 / 3 × 8 7 / 3 = 42 . 6 (c) Write down the diFerential of f at (8 , 64 , 2). Using the diFerential, approximate the change in f when x increases by 0.1 units, starting from (8 , 64 , 2).

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

View Full Document
2 Ans: f ( x, y, z ) = b yz xz yx B so that f (8 , 64 , 2) = b 128 16 512 B df = f ( x, y, z ) dx y p ( x ) dx z p ( x ) dx p = b 128 16 512 B 0 . 1 16 × 0 . 1 0 . 0833 × 0 . 1 = 12 . 8 + 16 × 0 . 16 + 512 × 0 . 025 = 42 . 7 (d) Identify the direction h * that ( x, y, z ) moves in, starting from (8 , 64 , 2), when x in- creases. Write down the directional derivative of f in the direction h * , i.e., f h * ( · , · , · ), and evaluate this derivative at (8 , 64 , 2). Using f h * (8 , 64 , 2), approximate the change in f when x increases by 0.1 units, starting from (8 , 64 , 2). Ans: When x increases by one, the vector ( x, y, z ) increases in the direction ( dx, y p ( x ) dx, z p ( x ) dx ) . When x = 8 and dx = 1 , therefore, ( x, y, z ) increases in the direction h * = (1 , 16 , 0 . 0833) . The unit length vector pointing in this direction is h * / || h * || = (0 . 0624 , 0 . 9980 , 0 . 0052) . Therefore, using the diFerential to compute the directional derivative, f h * (8 , 64 , 2) = b 128 16 512 B 0 . 0624 0 . 9980 0 . 0052 = 26 . 613 . Now a dx of 0.1 induces a shift in R 3 of 0 . 1 h * , which has length 0 . 1 || h * || . df = f h * (8 , 64 , 2) × 0 . 1 || h * || = 42 . 67 (e) Check to see that all four of these distinct methods give you the same answer! Ans: Amazingly, they do! (2) Recall that a function f : R n R m is nothing more than m functions, f 1 ...f m , each mapping R n R , and stacked on top of each other. (a) Using this fact, write down a formal deFnition of the directional derivative of f at x 0 in the direction h R n , for a function f : R n R m . Your deFnition should be of the
3 form blah, blah = lim k →∞ blah blah blah, blah Ans: Defnition: Given f : R n R m and h R n , the directional derivative oF f at x 0 in the direction h is given by, For i = 1 , ...m , f i h = lim | k |→∞ (

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.

## psCalculus2Ans - Fall 2007 Problem Set #10- Answer key...

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

View Full Document
Ask a homework question - tutors are online