whose domain D is a set of n-tuples in is defined. The range of f is the set of all values f x1,..., x n

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1 Grade 1 (MCV4UE) - AP Calculus Etended Page 1 o 1 A unction o n-variales is a real-valued unction 1,..., n whose domain D is a set o n-tuples 1,..., n in which 1,..., n is deined. The range o is the set o all values 1,..., n or 1,..., 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 1, 1 1,. The Eample 1: Domain and Range Sketch the domains o:, 9 a) ) g,, z lnz 1 z 1 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, Continuit: A unction, is continuous at a point P = lim, a,, a, a, in its domain i Online D Function Grapher: D Function Grapher:

2 Grade 1 (MCV4UE) - AP Calculus Etended Page o 1 Eample : Evaluating Limits Sustitutions Show, 1 is continuous. Evaluate lim,, 1,. Eample : Evaluating Limits Sustitutions Evaluate e 1 lim tan., 1,1 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, Eample 4: Evaluating Limits o Composite Functions Write H, e as a composite unction and evaluate lim H,, 1,, then the composite unction. Eample 5: Evaluating Limits at the origin Evaluate a) lim. ), 0,0 lim, 0, # #1

3 Grade 1 (MCV4UE) - AP Calculus Etended Page o Answers

4 Grade 1 (MCV4UE) - AP Calculus Etended Page 4 o Answers

5 Grade 1 (MCV4UE) - AP Calculus Etended Page 5 o 1 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 Leiniz notation o partial derivatives is and a, a,, a, I z,, then we also write a, z z and. Eample 1: Computing Partial derivatives, Compute the partial derivatives o 5 is the derivative at a,, as a unction o Eample : Computing Partial derivatives at particular values Calculate g 1, and g 1,, where g,. 1

6 Grade 1 (MCV4UE) - AP Calculus Etended Page 6 o 1 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 Eample 4: Computing Partial derivatives at particular values z w e Calculate z 0,1,0,1 where,, z, w. z w 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, Leiniz notation or the higher-order partial derivatives is

7 Grade 1 (MCV4UE) - AP Calculus Etended Page 7 o 1 Eample 5: Higher-Order Partial Derivatives Calculate the second-order partials and, e o 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 1 Check that or W PV T VT T V

8 Grade 1 (MCV4UE) - AP Calculus Etended Page 8 o 1 Eample 7: Choosing wisel when taking Higher-Order Mied Partial Derivatives Calculate the derivative g zzw, where g,, z, w w z sin z 15. #1-44

9 Grade 1 (MCV4UE) - AP Calculus Etended Page 9 o 1 Eercise 15. Answers

10 Grade 1 (MCV4UE) - AP Calculus Etended Page 10 o 1 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 z, at a, a, z L,. Eplicitl, z 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 Eample 9: Checking Dierentiailit h, is dierentiale. Check i

11 Grade 1 (MCV4UE) - AP Calculus Etended Page 11 o 1 Eample 10: Tangent Plane Find the tangent plane to the graph o, at,. Other Derivatives involving Multi-variales Gradient and Directional Derivatives P a, The gradient o a unction, at a point is the vector p a,, a, In three variales, i P a,, c, vector a,, c, a,, c, a, c p z, Eample 11: Gradient in Two variales Find the gradient o, P 1,1. at Eample 1: Gradient in Three variales g,, z z Find the gradient o #11-18

12 Grade 1 (MCV4UE) - AP Calculus Etended Page 1 o 1 Eercise 15.4 Answers