{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# w08mid - Math 237 Midterm Solutions Winter 2008 1 Short...

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

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.

Unformatted text preview: Math 237 - Midterm Solutions Winter 2008 1. Short Answer Problems [1] a) Let f : R 2 → R . Write the precise definition of f being differentiable at ( a, b ). If f x and f y both exist at ( a, b ) and lim ( x,y ) → ( a,b ) | R 1 , ( a,b ) ( x,y ) | bardbl ( x,y ) − ( a,b ) bardbl = 0 , where R 1 , ( a,b ) ( x, y ) = f ( x, y ) − L ( a,b ) ( x, y ), then f is differentiable at ( a, b ). [2] b) If f x and f y are both continuous at ( a, b ) what two things can we conclude about f ( x, y )? f is differentiable at ( a, b ) and f is continuous at ( a, b ). [1] c) What is the equation for the tangent plane to the surface 0 = f ( x, y, z ) at ( a, b, c )? ∇ f ( a, b, c ) · ( x − a, y − b, z − c ) = 0 . [1] d) State the definition of the directional derivative D u f ( a, b ) at a point ( a, b ) in the direction of a unit vector u . D u f ( a, b ) = d ds f ( ( a, b ) + s ( u 1 , u 2 ) ) vextendsingle vextendsingle vextendsingle vextendsingle s =0 . OR D u f ( a, b ) = d ds f ( a + su ) vextendsingle vextendsingle vextendsingle vextendsingle s =0 . OR D u f ( a, b ) = lim h → f ( a + hu ) − f ( a ) h . Math 237 - Midterm Solutions Winter 2008 2. Let f ( x, y ) = radicalbig 1 + x 2 − y 2 . [2] a) Sketch the domain of f . What is the range of f ? Solution: We must have 1 + x 2 − y 2 ≥ 0 hence we must have y 2 ≤ 1 + x 2 ⇒ | y | ≤ √ 1 + x 2 ⇒ − √ 1 + x 2 ≤ y ≤ √ 1 + x 2 . Which gives the diagram below. The range is clealy z ≥ 0. [3] b) Sketch the level curve 1 = f ( x, y ), the cross-section x = 1 and the cross section y = 1 of the surface z = f ( x, y ). Solution: Level Curves: z = 1 1 = radicalbig 1 + x 2 − y 2 1 = 1 + x 2 − y 2 x 2 = y 2 Cross sections: x = 1 z = radicalbig 1 + 1 2 − y 2 z = radicalbig 2 − y 2 Cross sections: y = 1 z = √ 1 + x 2 − 1 2 z = √ x 2 = | x | Math 237 - Midterm Solutions Winter 2008 3. Determine if each of the following limits exist. Evaluate the limits that exist.Determine if each of the following limits exist....
View Full Document

{[ snackBarMessage ]}

### Page1 / 8

w08mid - Math 237 Midterm Solutions Winter 2008 1 Short...

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

View Full Document
Ask a homework question - tutors are online