ESSENTIAL CALCULUS CH11 Partial derivatives

In this Chapter: 11.1 Functions of Several Variables 11.2 Limits and Continuity 11.3 Partial Derivatives 11.4 Tangent Planes and Linear Approximations 11.5 The Chain Rule 11.6 Directional Derivatives and the Gradient Vector 11.7 Maximum and Minimum Values 11.8 Lagrange Multipliers Review

DEFINITION A function f of two variables is a rule that assigns to each ordered pair of real numbers (x, y) in a set D a unique real number denoted by f (x, y). The set D is the domain of f and its range is the set of values that f takes on, that is, . Chapter 11, 11.1, P593

We often write z=f (x, y) to make explicit the value taken on by f at the general point (x, y) . The variables x and y are independent variables and z is the dependent variable. Chapter 11, 11.1, P593

Chapter 11, 11.1, P593

Domain of Chapter 11, 11.1, P594

Domain of Chapter 11, 11.1, P594

Domain of Chapter 11, 11.1, P594

DEFINITION If f is a function of two variables with domain D, then the graph of is the set of all points (x, y, z) in R3 such that z=f (x, y) and (x, y) is in D. Chapter 11, 11.1, P594

Chapter 11, 11.1, P595

Graph of Chapter 11, 11.1, P595

Graph of Chapter 11, 11.1, P595

Chapter 11, 11.1, P596

DEFINITION The level curves of a function f of two variables are the curves with equations f (x, y)=k, where k is a constant (in the range of f). Chapter 11, 11.1, P596

Chapter 11, 11.1, P597

Chapter 11, 11.1, P597

Chapter 11, 11.1, P598

Contour map of Chapter 11, 11.1, P598

Contour map of Chapter 11, 11.1, P598

is formed by lifting the level curves. The graph of h (x, y)=4x2+y2 is formed by lifting the level curves. Chapter 11, 11.1, P599

Chapter 11, 11.1, P599

Chapter 11, 11.1, P599

DEFINITION Let f be a function of two variables whose domain D includes points arbitrarily close to (a, b). Then we say that the limit of f (x, y) as (x, y) approaches (a ,b) is L and we write if for every number ε> 0 there is a corresponding number δ> 0 such that If and then Chapter 11, 11.2, P604

Chapter 11, 11.2, P604

Chapter 11, 11.2, P604

Chapter 11, 11.2, P604

If f( x, y)→L1 as (x, y)→ (a ,b) along a path C1 and f (x, y) →L2 as (x, y)→ (a, b) along a path C2, where L1≠L2, then lim (x, y)→ (a, b) f (x, y) does not exist. Chapter 11, 11.2, P605

4. DEFINITION A function f of two variables is called continuous at (a, b) if We say f is continuous on D if f is continuous at every point (a, b) in D. Chapter 11, 11.2, P607

5.If f is defined on a subset D of Rn, then lim x→a f(x) =L means that for every number ε> 0 there is a corresponding number δ> 0 such that If and then Chapter 11, 11.2, P609

4, If f is a function of two variables, its partial derivatives are the functions fx and fy defined by Chapter 11, 11.3, P611

NOTATIONS FOR PARTIAL DERIVATIVES If Z=f (x, y) , we write Chapter 11, 11.3, P612

RULE FOR FINDING PARTIAL DERIVATIVES OF z=f (x, y) To find fx, regard y as a constant and differentiate f (x, y) with respect to x. 2. To find fy, regard x as a constant and differentiate f (x, y) with respect to y. Chapter 11, 11.3, P612

The partial derivatives of f at (a, b) are FIGURE 1 The partial derivatives of f at (a, b) are the slopes of the tangents to C1 and C2. Chapter 11, 11.3, P612

Chapter 11, 11.3, P613

Chapter 11, 11.3, P613

The second partial derivatives of f The second partial derivatives of f. If z=f (x, y), we use the following notation: Chapter 11, 11.3, P614

CLAIRAUT’S THEOREM Suppose f is defined on a disk D that contains the point (a, b) . If the functions fxy and fyx are both continuous on D, then Chapter 11, 11.3, P615

The tangent plane contains the tangent lines T1 and T2 FIGURE 1 The tangent plane contains the tangent lines T1 and T2 Chapter 11, 11.4, P619

2. Suppose f has continuous partial derivatives 2. Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z=f (x, y) at the point P (xo ,yo ,zo) is Chapter 11, 11.4, P620

The linear function whose graph is this tangent plane, namely 3. is called the linearization of f at (a, b) and the approximation 4. is called the linear approximation or the tangent plane approximation of f at (a, b) Chapter 11, 11.4, P621

where ε1 and ε2→ 0 as (∆x, ∆y)→(0,0). 7. DEFINITION If z= f (x, y), then f is differentiable at (a, b) if ∆z can be expressed in the form where ε1 and ε2→ 0 as (∆x, ∆y)→(0,0). Chapter 11, 11.4, P622

8. THEOREM If the partial derivatives fx and fy exist near (a, b) and are continuous at (a, b), then f is differentiable at (a, b). Chapter 11, 11.4, P622

Chapter 11, 11.4, P623

For a differentiable function of two variables, z= f (x ,y), we define the differentials dx and dy to be independent variables; that is, they can be given any values. Then the differential dz, also called the total differential, is defined by Chapter 11, 11.4, P623

Chapter 11, 11.4, P624

For such functions the linear approximation is and the linearization L (x, y, z) is the right side of this expression. Chapter 11, 11.4, P625

If w=f (x, y, z), then the increment of w is The differential dw is defined in terms of the differentials dx, dy, and dz of the independent variables by Chapter 11, 11.4, P625

2. THE CHAIN RULE (CASE 1) Suppose that z=f (x, y) is a differentiable function of x and y, where x=g (t) and y=h (t) and are both differentiable functions of t. Then z is a differentiable function of t and Chapter 11, 11.5, P627

Chapter 11, 11.5, P628

3. THE CHAIN RULE (CASE 2) Suppose that z=f (x, y) is a differentiable function of x and y, where x=g (s, t) and y=h (s, t) are differentiable functions of s and t. Then Chapter 11, 11.5, P629

Chapter 11, 11.5, P630

Chapter 11, 11.5, P630

Chapter 11, 11.5, P630

4. THE CHAIN RULE (GENERAL VERSION) Suppose that u is a differentiable function of the n variables x1, x2,‧‧‧,xn and each xj is a differentiable function of the m variables t1, t2,‧‧‧,tm Then u is a function of t1, t2,‧‧‧, tm and for each i=1,2,‧‧‧,m. Chapter 11, 11.5, P630

Chapter 11, 11.5, P631

F (x, y)=0. Since both x and y are functions of x, we obtain But dx /dx=1, so if ∂F/∂y≠0 we solve for dy/dx and obtain Chapter 11, 11.5, P632

so this equation becomes F (x, y, z)=0 But and so this equation becomes If ∂F/∂z≠0 ,we solve for ∂z/∂x and obtain the first formula in Equations 7. The formula for ∂z/∂y is obtained in a similar manner. Chapter 11, 11.5, P632

Chapter 11, 11.6, P636

2. DEFINITION The directional derivative of f at (xo,yo) in the direction of a unit vector u=<a, b> is if this limit exists. Chapter 11, 11.6, P636

3. THEOREM If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector u=<a, b> and Chapter 11, 11.6, P637

8. DEFINITION If f is a function of two variables x and y , then the gradient of f is the vector function ∆f defined by Chapter 11, 11.6, P638

Chapter 11, 11.6, P638

10. DEFINITION The directional derivative of f at (x0, y0, z0) in the direction of a unit vector u=<a, b, c> is if this limit exists. Chapter 11, 11.6, P639

Chapter 11, 11.6, P639

Chapter 11, 11.6, P639

Chapter 11, 11.6, P640

15. THEOREM Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative Du f(x) is │▽f (x)│ and it occurs when u has the same direction as the gradient vector ▽ f(x) . Chapter 11, 11.6, P640

The equation of this tangent plane as The symmetric equations of the normal line to soot P are Chapter 11, 11.6, P642

Chapter 11, 11.6, P644

Chapter 11, 11.6, P644

Chapter 11, 11.7, P647

1. DEFINITION A function of two variables has a local maximum at (a, b) if f (x, y) ≤ f (a, b) when (x, y) is near (a, b). [This means that f (x, y) ≤ f (a, b) for all points (x, y) in some disk with center (a, b).] The number f (a, b) is called a local maximum value. If f (x, y) ≥ f (a, b) when (x, y) is near (a, b), then f (a, b) is a local minimum value. Chapter 11, 11.7, P647

2. THEOREM If f has a local maximum or minimum at (a, b) and the first order partial derivatives of f exist there, then fx(a, b)=1 and fy(a, b)=0. Chapter 11, 11.7, P647

A point (a, b) is called a critical point (or stationary point) of f if fx (a, b)=0 and fy (a, b)=0, or if one of these partial derivatives does not exist. Chapter 11, 11.7, P647

Chapter 11, 11.7, P648

If D>0 and fxx (a, b)>0 , then f (a, b) is a local minimum. 3. SECOND DERIVATIVES TEST Suppose the second partial derivatives of f are continuous on a disk with center (a, b) , and suppose that fx (a, b) and fy (a, b)=0 [that is, (a, b) is a critical point of f]. Let If D>0 and fxx (a, b)>0 , then f (a, b) is a local minimum. (b)If D>0 and fxx (a, b)<0, then f (a, b) is a local maximum. (c) If D<0, then f (a, b) is not a local maximum or minimum. Chapter 11, 11.7, P648

NOTE 1 In case (c) the point (a, b) is called a saddle point of f and the graph of f crosses its tangent plane at (a, b). NOTE 2 If D=0, the test gives no information: f could have a local maximum or local minimum at (a, b), or (a, b) could be a saddle point of f. NOTE 3 To remember the formula for D it’s helpful to write it as a determinant: Chapter 11, 11.7, P648

Chapter 11, 11.7, P649

Chapter 11, 11.7, P649

Chapter 11, 11.7, P651

4. EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES If f is continuous on a closed, bounded set D in R2, then f attains an absolute maximum value f(x1,y1) and an absolute minimum value f(x2,y2) at some points (x1,y1) and (x2,y2) in D. Chapter 11, 11.7, P651

1. Find the values of f at the critical points of in D. 5. To find the absolute maximum and minimum values of a continuous function f on a closed, bounded set D: 1. Find the values of f at the critical points of in D. 2. Find the extreme values of f on the boundary of D. 3. The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value. Chapter 11, 11.7, P651

Chapter 11, 11.7, P652

Chapter 11, 11.8, P654

Chapter 11, 11.8, P655

(a) Find all values of x, y, z, and such that METHOD OF LAGRANGE MULTIPLIERS To find the maximum and minimum values of f (x, y, z) subject to the constraint g (x, y, z)=k [assuming that these extreme values exist and ▽g≠0 on the surface g (x, y, z)=k]: (a) Find all values of x, y, z, and such that and (b) Evaluate f at all the points (x, y, z) that result from step (a). The largest of these values is the maximum value of f; the smallest is the minimum value of f. Chapter 11, 11.8, P655

Chapter 11, 11.8, P657

Chapter 11, 11.8, P657