Grade (MCV4UE) - AP Calculus Etended Page o A unction o n-variales is a real-valued unction... n whose domain D is a set o n-tuples... n in which... n is deined. The range o is the set o all values... n or... n in the domain. For instance e has domain D R since is deined or all range is R. n R. When is deined a ormula we usuall take as domain the set o all n-tuples or. The Eample : Domain and Range Sketch the domains o: 9 a) D 9 : : 9 is deined when 9 0 a disk o radius centered at the origin. ) g ln g ln is deined i oth and ln D : : 0 are deined. Eample : Graphing Multi-variale Function 6 Descrie the image o the unction. 6 or 6 0 which represents a plane with n =. intercept at = intercept at = intercept at = 6 6 Limit Laws or Multivariale unctions lim lim g Assume that and a Then: (i) Sum Law: lim g a a = lim + lim g a (iii) Product Law: lim g a = a a lim lim g a eist. (ii) Constant Multiple Law: lim k = k lim a (v) Quotient Law: lim g a a I 0 a lim lim = g a lim g a
Grade (MCV4UE) - AP Calculus Etended Page o Continuit: A unction is continuous at a point P = lim a a a in its domain i Online D Function Grapher: http://www.livephsics.com/tools/mathematical-tools/online--d-unction-grapher/ D Function Grapher: http://www.math.uri.edu/~kaskos/lashmo/graphd/ Eample : Evaluating Limits Sustitutions Show is continuous. Evaluate lim lim Thereore is continuous. Eample : Evaluating Limits Sustitutions Evaluate e lim tan. lim e tan = 4 4 6 lim e lim tan = e tan = e 4 4e. A Composite o Continuous unctions is Continuous I is continuous at a and u a. G is continuous at G is continuous at c a then the composite unction Eample 4: Evaluating Limits o Composite Functions Write H e as a composite unction and evaluate lim H u G where Gu e and H limit using sustitution: lim H = = lim e e = e.. Since G and are continuous we ma evaluate the
Grade (MCV4UE) - AP Calculus Etended Page o Eample 5: Evaluating Limits at the origin Evaluate a) lim. ) 00 lim. 00 a) 0 lim (Indetermined result) 00 0 Limit along -ais = lim 0 lim lim Limit along -ais = lim 0 lim lim 0 0 0 0 0 These two limits are dierent and hence lim ) 00 0 00 0 lim (Indetermined result) 0 lim 0 lim lim 0 0 0 0 does not eist. 0 0 0 0 Limit along -ais = 0 Limit along -ais = lim 0 lim lim 0 0 These two limits are same and hence lim 00 eists = 0. 0 0 0 0 0 0 5. 5.# 9 5. # Answers
Grade (MCV4UE) - AP Calculus Etended Page 4 o 5. Answers
Grade (MCV4UE) - AP Calculus Etended Page 5 o Partial Derivatives In calculus o one variale the derivative ' a is the rate o change o at a. B contrast a unction o two or more variales does not have a unique rate o change ecause each variale ma aect in dierent was. The partial derivatives are the rates o change with respect to each variale separatel. A unction o two variales has two partial derivatives denoted and deined the ollowing limits (i the eist). a h a a lim h a k a a lim 0 h k 0 k is the derivative at as a unction o alone and alone. The Leini notation o partial derivatives is and a a a I then we also write a and. Eample : Computing Partial derivatives 5 5 5 5 Compute the partial derivatives o 5 is the derivative at a as a unction o 5 5 4 4 5 5 Eample : Computing Partial derivatives at particular values Calculate g and g where g. g To calculate treat (and thereore ) as a constant: 6 g g To calculate g g 6 4 54 7 6 8 4 g treat (and thereore ) as a constant: 6 8 4
Grade (MCV4UE) - AP Calculus Etended Page 6 o Chain Rule in Partial Derivatives We use the Chain Rule in the usual wa to compute the partial derivative o a composite unction Fg where F u is a unction o one variale and u g : df du u df du u Eample : Chain Rule in Partial Derivatives Compute sin F u sin Let u F sin u and u df u sin cos du cos Eample 4: Computing Partial derivatives at particular values w e Calculate 00 where w. w w w w w e e w e w w w w w we e w 00 0 0 0 0 e e 0 = e Higher-Order Partial Derivatives The higher-order partial derivatives o a unction. For instance the second-order partial derivatives are the derivatives o We write or derivative o and are the derivatives o the partial derivatives and and with respect to or. or derivative o. We can also have the mied partial which are the derivatives Leini notation or the higher-order partial derivatives is
Grade (MCV4UE) - AP Calculus Etended Page 7 o Eample 5: Higher-Order Partial Derivatives Calculate the second-order partials and e o e e e e e 6 e e e e e e e e e Equalit o Mied Partials I and are oth continuous unctions on a disk D then a a or all a D In other words Eample 6: Higher-Order Mied Partial Derivatives 4 W W Check that or W PV T VT T V 4 W W PV T 4 PV T T V 4 W W W PV T 4 PV T T T T TV T V 8 W W W PV T PV T VT V T T V T TV W W 8 Eample 7: Choosing wisel when taking Higher-Order Mied Partial Derivatives Calculate the derivative g w where g w w sin Since derivatives ma e calculated in an order to dierentiate with respect to w irst. This cause the second term which does not depend on w to disappear: g w w sin w w w Net dierentiate twice with respect to and once with respect to : g w w w 4 w g w 4 w 4 w 4 w g w 4 w w We conclude that g g w 5. # - 44 w w.
Grade (MCV4UE) - AP Calculus Etended Page 8 o Eercise 5. Answers
Grade (MCV4UE) - AP Calculus Etended Page 9 o Dierentiailit and the Tangent Plane Assume that i: a a eist and is deined in a disk D containing a. We sa that is dierentiale at a and is locall linear at a In this case the tangent plane to the surace at a a L. Eplicitl a a a a I is dierentiale at all points in a domain D which means and is dierentiale on D. is the plane with equation eist we sa that Eample 8: Checking Dierentiailit sin is dierentiale. Show that sin cos 0sin cos sin cos cos Since the partial derivatives are continuous thereore the unction is dierentiale. Eample 9: Checking Dierentiailit h is dierentiale. Check i h h The unction h is onl dierentiale or 00 continuous ecept at 0 0. h is not dierentiale at 0 0 h is not dierentiale at 0 0 ecause the partial derivatives eist and are
Grade (MCV4UE) - AP Calculus Etended Page 0 o Eample 0: Tangent Plane Find the tangent plane to the graph o at 8 4 4 64 4 Since 8 4 thereore the tangent plane has equation: 4 4 4 8 4 48 44 4 4 4 4 44 0 (Scalar equation o plane). Recall: Equation o Tangent Plane to at a a a a a a : Other Derivatives involving Multi-variales Gradient and Directional Derivatives The directional derivative o at 0 0 in the direction o a unit vector u a is 0 ha 0 h 0 0 Du 0 0 lim i this limit eists. h 0 h I is a dierentiale unction o and then has a directional derivative in the direction o an unit vector u a and D a u I the unit vector u makes an angle with the positive -ais then we can write u cos sin and the ormula ecomes Du cos sin The gradient o a unction at a point P a is the vector p a a In three variales i P a c vector a c a c a c p Eample : Directional Derivative Find the directional derivative D u i 4 6. What is D u D u? cos 8 8 6 sin 6 and u is the unit vector given angle Thereore: D u 8
Grade (MCV4UE) - AP Calculus Etended Page o Eample : Gradient in Two variales Find the gradient o at P. So At P p Eample : Gradient in Three variales Find the gradient o g 8 g 8 8 7 7 8 7 6 5.4 #-8 Eercise 5.4 Answers