Midterm Review Guide
Midterm Review Guide MA1024
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Date Created: 02/07/15
Calculus 4 Midterm Basics 141 Functions of Several Variables Level curves to draw a level curve 2D 0 If fxyxy1 choose different c values 0 c 1 O 1 9 set equation equal to c value 0 Graph equation and label it with its corresponding c value The closer together the lines on the level curve the steeper the 3D object is Level surfaces 3D objects 0 If fxy x2y2 9 z x2y2 0 Fix xxo yyo and 220 in which xoyo and 20 are all constants 0 Bring constants to one side then figure out its 2D shape 2D shapes 0 yZzxo2 9 parabola o x2y2 z0 9 circle 0 9y222 16 9 ellipse o 22169yo2 9 two lines positive amp negative 0 22y2 x0 9 hyperbola 142 Limits and Continuity 0 Limits limxyoo Jag 7 to figure out if this has a limit use these different methods 0 Substitute ymx line in the equation If m slope is left after simplifying then it does not have a limit because it differs when you substitute other equations in with different slopes ex Y2x 0 Substitute xaxis in which y0 or yaxis in which x0 0 Substitute a line ymx and a parabola yx2 to yield different results 0 Use sandwich theorem 39 limtxyo0 ysin 1X I 1 S sin1XS 1 I thus y S ysin1x S y I If the xy approach 00 then limit of ysin1x approaches 0 143 Partial Derivatives Partial derivative Differentiating one variable while the other is held constant 1 if 62f 6yax ayax 9 fxy a a f 62 f 9f 0 6x 6x 6x242 XX 6 6f 6 f o 9f 6x6y axay yx 11f 62f 6y6y ayAz 9f 146 Tangent Planes To find a tangent plane to a function use equation axxo by yo cz 20 O at point Pxo yo 20 0 Vector lta b cgt s normal vector of plane a a i X0 yo 20 a b X0 yo 20 C 3 x0 yo 20 Linearization You can do this because Lxy is a good approximation of fxy at the given point 0 LOW f W fxXo yoX Xo f Xo yo yyo 145 Gradients and Directional Derivatives Gradient of f vector 2 ltfX x0 yo fyxo y0gt Directional Derivative How fast a function is changing o 66 Duf fu gradient f I u 9 dot product gives numerical value not vector u direction gradient f magnitude W 0 Increase most rapidly is u o Decrease most rapidly is u vector value O 147 Extreme Values and Saddle Points Critical point is an interior point when 0 f x 0 o f x DNE Theorem If fx has a local min or max value at an interior point x of its domain then x is a critical point 0 However every critical point is not a min max value Theorem If fxy is a local min or max value at an interior point xy then xy is a critical point 0 If x is a critical point I fquot x gt O 9 local min I fquot x lt O 9 local max I fquot x O 9 inconclusive Let fxxo yo fy x0 yo 0 o A fXX x0 yo 0 B fxytxm yo 0 C fyy X0 yo 0 D AC B2 9 discriminant 1 IfDgt OandAgt O 9localmin DgtO and A lt0 9 local max 2 If Dlt 0 saddle point if you look at function from different angles the point can be either a maximum or minimum 3 If D O inconclusive Global Min Max To find a global minmax you must test critical points and boundaries 0 Find critical points and plug into original equation 0 Find boundaries and critical points from all of the function s segments then plug into original equation I Critical points are found by taking partial derivatives and setting them equal to O 0 Compare the points and see which value is highest and at which points
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