MATH 251 Midterm 2 — What You Must Memorize (Learn Mode)

14.3: What is the limit definition of the partial derivative f_x(x, y)?

lim (h->0) [f(x+h, y) - f(x, y)] / h

14.3: How do you evaluate a partial derivative with respect to x?

Differentiate with respect to x while treating y as a constant.

14.3: What is the standard notation for the partial derivative of f with respect to y?

f_y, (∂f/∂y), or (∂z/∂y)

14.3: What is the geometric interpretation of the partial derivative f_x(a, b)?

The slope of the tangent line to the surface curve at (a, b) in the plane y = b.

14.3: Under what condition do mixed partials like f_xy and f_yx equal each other?

When both mixed partials are continuous on an open disk containing the point.

14.3: What differential approximation formula estimates the change dz given dx and dy?

dz ≈ f_x dx + f_y dy

14.4: What is the single-variable chain rule formula for w = f(x(t), y(t))?

dw/dt = (∂w/∂x)(dx/dt) + (∂w/∂y)(dy/dt)

14.4: What is the chain rule formula for w = f(x, y, z) where x, y, and z depend on t?

dw/dt = (∂w/∂x)(dx/dt) + (∂w/∂y)(dy/dt) + (∂w/∂z)(dz/dt)

14.4: In the chain rule for w = f(x(t), y(t)), what do the partial derivatives represent?

The rates of change of f with respect to its intermediate variables x and y.

14.4: What is ∂w/∂r when w = f(x, y) and both x and y depend on r and s?

(∂w/∂x)(∂x/∂r) + (∂w/∂y)(∂y/∂r)

14.4: What is ∂w/∂s when w = f(x, y) and both x and y depend on r and s?

(∂w/∂x)(∂x/∂s) + (∂w/∂y)(∂y/∂s)

14.5: What are the components of the gradient vector ∇f in 2D?

14.5: What are the components of the gradient vector ∇f in 3D?

14.5: How is the directional derivative D_u f computed using the gradient vector?

D_u f = ∇f · u

14.5: What condition must unit vector u satisfy in the directional derivative formula?

Its length must equal 1 (||u|| = 1).

14.5: In what direction does a function f increase most rapidly at a given point?

In the direction of the gradient vector ∇f.

14.5: What is the maximum value of the directional derivative D_u f at any point?

The magnitude of the gradient, ||∇f||.

14.6: What condition guarantees that a function f(x, y) is differentiable at (a, b)?

The partial derivatives f_x and f_y exist and are continuous near (a, b).

14.6: What is the equation of the tangent plane to the surface z = f(x, y) at (x_0, y_0, z_0)?

z - z_0 = f_x(x - x_0) + f_y(y - y_0)

14.6: What is the equation of the tangent plane to a level surface F(x, y, z) = c at a point?

F_x(x - x_0) + F_y(y - y_0) + F_z(z - z_0) = 0

14.6: What is the direction vector for the normal line to a level surface F(x, y, z) = c?

The gradient vector

14.6: What is the linearization L(x, y) of a function f(x, y) at (a, b)?

f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

14.6: What requirement must the gradient satisfy for an implicit surface F(x, y, z) = c to be regular?

The gradient ∇F must be non-zero everywhere on the surface.

14.7: What defines a critical point (a, b) of a function f(x, y)?

Both f_x(a, b) = 0 and f_y(a, b) = 0, or one/both partial derivatives do not exist.

14.7: What are the necessary conditions for an interior local extremum of a differentiable function?

The gradient ∇f must equal the zero vector

14.7: What does a discriminant value D > 0 and f_xx < 0 indicate in the second derivative test?

A local maximum.

14.7: What does a discriminant value D > 0 and f_xx > 0 indicate in the second derivative test?

A local minimum.

14.7: What does a negative discriminant (D < 0) indicate in the second derivative test?

A saddle point.

14.7: What conclusion can be drawn if the second derivative test discriminant equals zero (D = 0)?

The test is inconclusive.

14.7: What two sets of points must be checked to find absolute extrema on a closed bounded region R?

Interior critical points and boundary points.

14.8: How do you set up the system of equations for optimization with a constraint g(x, y) = c?

Set ∇f = λ∇g and include the constraint equation g(x, y) = c.

14.8: Why must the gradient vectors ∇f and ∇g be parallel at an extremum subject to g(x, y) = c?

Level curves of f and g must be tangent, meaning their normal vectors are scalar multiples.

14.8: After solving the Lagrange multiplier system for candidate points, how do you find absolute extrema?

Evaluate f(x, y) at all candidate points and select the largest and smallest values.

14.8: What non-zero condition must the constraint gradient satisfy at the solution point?

The gradient ∇g must not be the zero vector

14.8: What does the Lagrange multiplier λ itself represent conceptually?

The instantaneous rate of change of the maximum value with respect to the constraint constant c.

15.1: How is a double integral over a rectangle defined using Riemann sums?

As the limit of double Riemann sums as the subrectangle dimensions approach zero.

15.1: What is the geometric interpretation of a double integral ∬_R f(x, y) dA when f(x, y) ≥ 0?

The volume of the solid lying under the surface z = f(x, y) and above the region R.

15.1: What theorem allows a double integral over a rectangle to be evaluated as an iterated integral?

Fubini's Theorem for Rectangles.

15.1: What continuity condition is required for Fubini's Theorem to apply to a double integral?

The function f(x, y) must be continuous on the rectangular region R.

15.1: What does a double integral ∬_R 1 dA evaluate to geometrically?

The area of the region R.

15.2: In an iterated integral where dx is inside and dy is outside, which variable is integrated first?

The inner variable x.

15.2: When evaluating the inner integral with respect to x, how are variables of y treated?

As constants.

15.2: What is the result of evaluating an iterated integral with respect to both variables sequentially?

A single numerical value representing volume or cumulative quantity.

15.2: Does the order of integration matter for a continuous function over a rectangular region?

No, Fubini's theorem guarantees iterated integrals in either order are equal.

15.2: What type of limits of integration are required for the outer integral in an iterated integral?

Constant numbers.

15.3: What defines a Type I region in the xy-plane?

A region bounded by two vertical lines and two continuous functions of x.

15.3: What are the inner bounds (limits of integration) for a Type I region iterated integral?

Functions of x, written as lower to upper y-bounds.

15.3: What defines a Type II region in the xy-plane?

A region bounded by two horizontal lines and two continuous functions of y.

15.3: How do you reverse the order of integration for a double integral over a general region?

Sketch the region, determine new inequalities, and rewrite the boundary limits.

15.3: How is the average value of a function f(x, y) over a region R calculated?

(1 / Area(R)) ∬_R f(x, y) dA

15.4: When is it advantageous to convert a double integral from rectangular to polar coordinates?

When the region of integration involves circles, disks, or sectors.

15.4: What formula relates polar radius r to rectangular coordinates x and y?

r^2 = x^2 + y^2

15.4: What extra factor must be included in the area element dA when switching to polar coordinates?

r (so dA = r dr dθ)

15.4: How do the limits of integration for r typically appear in a polar double integral setup?

From r_1(θ) to r_2(θ), representing radial bounds.

15.4: What are the standard inner and outer integration variables in a polar double integral?

r is integrated first (inner), and θ is integrated second (outer).