Calculus 3 Final Exam Review SheetPage 1 of 28
Section 11.1: Vector-Valued Functions
Definition 1.1: Vector-Valued Function
A vector-valued functionr(t) is a mapping from its domain to its range , so that for each t in D, r(t) = v for exactly one vV3. We can write a vector-valued function as
r(t) = f(t)i + g(t)j + h(t)k,
for some scalar functions f, g, and h (called the components of r).
The vector function
corresponds to the set of parametric equations
Arc length of a curve defined by a vector-valued functionr(t) = f(t)i + g(t)j + h(t)k, on an interval
t [a, b]:
Parametric equations for an intersection of surfaces:
- Eliminate one variable
- Set another variable to the parameter t.
- Write all variables in terms of t.
Section 11.2: The Calculus of Vector-Valued Functions
Definition 2.1: Definition of the Limit of a Vector-Valued Function
Given a vector-valued function , we define the limit of as t approaches aas follows:
,
provided all three of the above limits exist. If any one of the limits does not exist, then does not exist.
Definition 2.2: Continuity of a Vector-Valued Function
We say the vector-valued function is continuous at t = aif
.
Theorem 2.1: Condition for the Continuity of a Vector-Valued Function
A vector-valued function is continuous at t = aif and only if all of f, g, and h are continuous at t = a.
Definition 2.3: Definition of the Derivative of a Vector-Valued Function
The derivative of the vector-valued function is defined by
at any value of t for which this limit exists. If this limit exists at some value t = a, we say r is differentiable at a.
Theorem 2.2: Derivative of a Vector-Valued Function in Terms of its Components
If , and the components f, g, and h are all differentiable for some value of t, then r is also differentiable at that value of t and its derivative is given by .
Theorem 2.3: Derivatives of Combinations of Vector-Valued Functions and Scalars
Suppose and are differentiable vector-valued functions, is a differentiable scalar function, and c is a constant. Then
(i)
(ii)
(iii)
(iv)
(v)
Theorem 2.4: Orthogonality of a Constant-Magnitude Vector-Valued Function and Its Derivative
is constant throughout some interval if and only if r(t) and r(t) are orthogonal for all t in that interval.
Definition 2.4: Antiderivative of a Vector-Valued Function
The vector-valued function R(t) is an antiderivative of the vector-valued function r(t) if
R(t) = r(t).
Definition 2.5: Indefinite Integral of a Vector-Valued Function
If R(t) is any antiderivative of the vector-valued function r(t), then the indefinite integral of r(t) is defined as
,
where is an arbitrary constant vector.
Note that this simply corresponds to
Definition 2.6: Definite Integral of a Vector-Valued Function
For the vector-valued function , the definite integral of r(t) on the interval [a, b] is defined as:
,
provided all three integrals on the right exist.
Theorem 2.5: Extension of the Fundamental Theorem of Calculus to Vector-Valued Functions
If R(t) is an antiderivative of a vector-valued function r(t) on the interval [a, b], then:
.
Section 11.3: Motion in Space
r(t) = v(t), the velocity vector.
v(t) = r(t) = a(t), the acceleration vector.
Newton’s Second Law states that F = ma.
The general equations for a projectile launched with an initial velocity v0 at an angle of inclination from a location (x0, y0) are:
Time to reach maximum altitude:
Maximum altitude:
Range: Find the time to reach the ground, then use that time to calculate horizontal distance.
Section 11.4: Curvature
Unit Tangent Vector
Curvature
(nearly impossible to calculate this way)
(this is easier to work with, but still pretty hard)
(this is the easiest to compute)
For a 2D plane curve defined by the Cartesian function y = f(x),
For a 2D plane curve defined by the polar function r = f(),
Section 11.5: Tangent and Normal Vectors
The Unit Tangent Vector:
The Principal Unit Normal Vector (Definition 5.1):
(Note: N(t)always points toward the concave side of the curve.)
The Binormal Vector (Definition 5.2)
The osculating circle . . . .
- has the same curvature as the curve itself at a point P
- has the same tangent vector as the curve at P (namely,T is tangent to both).
- has a radius of , called the radius of curvature for the curve at P.
- is considered the circle of “best fit” for the curve at P.
Tangential and Normal Components of Acceleration
,,
Section 11.6: Parametric Surfaces
For any cylindrical surface formed by “drawing out” a curve defined by x = cos(t), y = sin(t) along the z axis:
x = cos(u), y = sin(u), z = v
Sphere centered at origin:
x2 + y2 + z2 = r2x = rcos(u)sin(v), y = rsin(u)sin(v), z = rcos(v)
2 general tricks for converting quadric surface equations to parametric equations:
cos2(u) + sin2(u) = 1
cosh2(u) – sinh2(u) = 1
Section 12.1: Functions of Several Variables
Definition: Function of Two Variables
A function of two variables is a rule that assigns a single real number to each ordered pair of numbers (x, y) in the domain of the function.
Definition: Function of Three Variables
A function of three variables is a rule that assigns a single real number to each ordered triple of numbers (x, y, z) in the domain of the function.
- Example: w = f(x, y, z) represents a surface embedded in 4 dimensions…
Contour Plots & Level Curves
Contour plots are 2-dimensional images (drawn in the xy-plane) that provide valuable information about a surface that lives in 3D space. A contour plot consists of a number of level curves for the surface. Alevel curve of is the (2D) graph of the curve , for some constant c. A contour plot is a number of level curves plotted on the same set of axes.
Section 12.2: Limits and Continuity
Definition 2.1: Formal Definition of Limit for a Function of Two Variables
Let f be defined on the interior of a circle centered at the point (a, b), except possibly at (a, b) itself. We say that if for every > 0 there exists a > 0 such that whenever .
Theorem 2.1 (Squeeze Theorem)
Suppose that for all (x, y) in the interior of some circle centered at (a, b), except possibly at (a, b). If , then .
Definition 2.2: Continuity of a Function of Two Variables
Suppose f(x, y) is defined in the interior of a circle centered at the point (a, b). We say that f is continuous at (a, b) if . If f(x, y) is not continuous at (a, b), then we call
(a, b) a discontinuity of f.
Theorem 2.2 (Continuity of a Composition of Continuous Functions)
Suppose that f(x, y) is continuous at (a, b), and g(x) is continuous at the point f(a, b). Then is continuous at (a, b).
Definition 2.3: Formal Definition of Limit for a Function of Three Variables
Let f(x, y, z) be defined on the interior of a circle centered at the point (a, b, c) except possibly at
(a, b, c) itself. We say that if for every > 0 there exists a > 0 such that whenever .
Definition 2.3: Continuity of a Function of Three Variables
Suppose f(x, y) is defined in the interior of a sphere centered at the point (a, b, c). We say that f is continuous at (a, b, c) if . If f(x, y, z) is not continuous at (a, b, c), then we call (a, b, c) a discontinuity of f.
Section 12.3: Partial Derivatives
Definition 3.1: Partial Derivative
The partial derivative of f(x, y) with respect to x, written as , is defined by
for any values of x and y for which the limit exists.
The partial derivative of f(x, y) with respect to y, written as , is defined by
for any values of x and y for which the limit exists.
Theorem 3.1 (Equality of Mixed Second-Order Partial Derivatives)
If fxy(x, y) and fyx(x, y) are continuous on an open set containing (a, b), then fxy(x, y) = fyx(x, y).
Section 12.4: Tangent Planes and Linear Approximations
Theorem 4.1: Equation of a Plane Tangent to a Surface
Suppose that f(x, y) has continuous first partial derivatives at (a, b). A normal vector to the tangent plane at z = f(x, y) at (a, b) is then <fx(a, b), fy(a, b), -1>. Further, an equation of the tangent plane is given by:
z = f(a, b) + fx(a, b)(x – a) + fy(a, b)(y – b).
Also, the equation of the normal line through the point (a, b, f(a, b)) is given by:
x = a + fx(a, b)t, y = b + fy(a, b)t, z = f(a, b) – t.
Theorem 4.2: Increment for a Function of Two Variables
Suppose that z = f(x, y) is defined on the rectangular region R = {(x, y) | x0xx1 and y0yy1} and fx and fy are defined on R and are continuous at (a, b) R. Then for (a + x, b + y) R,
z = fx(a, b)x + fy(a, b)y + 1x + 2y,
where 1 and 2are functions of x and y that both tend to zero as x and y tend to zero.
I.E., ,
where
Total Differential of a Function of Two Variables
Definition 4.1: Differentiability of a Function of Two Variables
Let z = f(x, y). We say that z is differentiable at (a, b) if we can write
z = fx(a, b)x + fy(a, b)y + 1x + 2y,
where 1 and 2are functions of both x and y and 1 and 2 0 as (x, y) 0. We say that f is differentiable on a region RR2 whenever f is differentiable at every point in R.
Definition 4.2: Linear Approximation for a Function of Three Variables
The linear approximation to at the point is given by:
Section 12.5: The Chain Rule
Theorem 5.1: The Chain Rule
If , where x(t) and y(t) are differentiable and f(x, y) is a differentiable function of x and y, then
Theorem 5.2: The Chain Rule
If , where f(x, y) is a differentiable function of x and y, and where and both have first-order partial derivatives, then the following chain rules apply:
and
Implicit Function Theorem
If , then , , etc.
Section 12.6: The Gradient and Directional Derivatives
Definition 6.1: The Directional Derivative
The directional derivative of f(x, y) at the point (a, b) and in the direction of is given by:
Theorem 6.1: The Directional Derivative as a Dot Product
If f(x, y) is differentiable at the point (a, b) and is any unit vector, then we can write:
Definition 6.2: The Gradient Vector
The gradient of f(x, y) is the vector-valued function
,
provided both partial derivatives exist.
Theorem 6.2: The Directional Derivative in Terms of the Gradient
If f(x, y) is differentiable and is any unit vector, then:
Theorem 6.3: The Directional Derivative in Terms of the Gradient
If f(x, y) is differentiable at the point (a, b), then:
(i).The maximum rate of change of f at (a, b) is , occurring in the direction of the gradient.
(ii).The minimum rate of change of f at (a, b) is -, occurring in the direction opposite of the gradient.
(iii).The rate of change of f at (a, b) is 0 in directions orthogonal to the gradient.
(iv).The gradient is orthogonal to the level curve f(x, y) = f(a, b).
Definition 6.3: The Gradient Vector for a 3 Variable Function
The directional derivative of f(x, y, z) at the point (a, b, c) and in the direction of is given by:
The gradient of f(x, y, z) is the vector-valued function
.
Definition 6.4: The Directional Derivative in Terms of the Gradient Vector
Theorem 6.5: The Directional Derivative in Terms of the Gradient Vector
If the point lies on the surface defined by , and the three partial derivatives , , and all exist at that point, then the vector is normal to the surface at that point, and the tangent plane is given by the equation
Section 12.7: Extrema of Functions of Several Variables
Definition 7.1: Local Extrema
We call f(a, b) a local maximum of f if there is an open disk R cantered at point (a, b) for which f(a, b) f(x, y) for all (x, y) R. Similarly, we call f(a, b) a local minimum of f if there is an open disk R cantered at point (a, b) for which f(a, b) f(x, y) for all (x, y) R. In either case,
f(a, b) is called a local extremum.
Definition 7.2: Critical Point
The point (a, b) is a critical point of the function f(x, y) if (a, b) is in the domain of f and either or one or both of and do not exist at (a, b).
Theorem7.1: Condition for a Local Extremum
If f(x, y) has a local extremum at (a, b), then (a, b) must be a critical point of f.
Definition 7.3: Saddle Point
The point P(a, b, f(a, b)) is a saddle point of z = f(x, y) if (a, b) is a critical point of f and if every open disk centered at (a, b) contains points (x, y) in the domain of f for which f(x, y) < f(a, b) and points
(x, y) in the domain of f for which f(x, y) > f(a, b).
Theorem7.2: Second Derivatives Test
Suppose that f(x, y) has continuous second-order partial derivatives in some open disk containing the point (a, b) and that fx(a, b) = fy(a, b) = 0. Define the discriminant D for the point (a, b) by:
D(a, b) = fxx(a, b)fyy(a, b) – [fxy(a, b)]2.
(i).If D(a, b) > 0 and fxx(a, b) > 0, then f has a local minimum at (a, b).
(ii).If D(a, b) > 0 and fxx(a, b) < 0, then f has a local maximum at (a, b).
(iii).If D(a, b) < 0, then f has a saddle point at (a, b).
(iv).If D(a, b) = 0, then no conclusion can be drawn.
Linear regression, by the technique of least squares: find the absolute minimum of a certain function f (the square of the residuals) of the two variables m and b.
You can show that, in general, for n data points (x1, y1), (x2, y2), …, (xn, yn), the linear least square fit yields the two equations
Which have solution
Method of Steepest ascent to find a local extrema of z = f(x, y):
- Pick a starting guess.
- “Move away” from in the direction of until you find coordinates such that
- Repeat step 2 until you are close enough.
Definition 7.4: Absolute Extrema
We call f(a, b) the absolute maximum of f if f(a, b) f(x, y) for all (x, y) domain. Similarly, we call f(a, b) the absolute minimum of f if f(a, b) f(x, y) for all (x, y) domain.
Theorem7.3: Extreme Value Theorem
Suppose that f(x, y) is continuous on a closed and bounded region . The f has both an absolute maximum and absolute minimum on R.
Section 12.8: Constrained Optimization and Lagrange Multipliers
The method of Lagrange Multipliers:
Theorem 8.1:
If and are functions with continuous first partial derivatives and on the surface and either the minimum or maximum value of subject to the constraint occurs at , then
for some constant (called the Lagrange multiplier).
Optimization with two constraints:
To optimize the function subject to the constraints and , solve
Three-variable constraint problem:
The profit a company makes on producing x, y, and z thousand units of products is given by:
P(x, y, z) = 4x + 8y + 6z. If manufacturing constraints force . Find the production parameters for maximum profit for the company.
Section 13.1: Double Integrals
Definition 1.1: The Definite Integral of a Function of a Single Variable
For any function f defined on the interval [a, b] and ||P|| (the norm of the partition) defined as the maximum of all the intervals on [a, b] (i.e., ||P|| = max{xi}), the definite integral of f on [a, b] is:
,
provided the limit exists and is the same for all values of the evaluation points ci [xi-1, xi] for
i = 1, 2, …, n. In this case, we say f is integrable on [a, b].
Riemann Sum Over a Region:
Definition 1.2: Double Integral of a Function of Two Variables Over a Rectangular Region
For any function f(x, y) defined on the rectangle R = {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d}, and ||P|| (the norm of the partition) defined as the maximum diagonal of any rectangle in the partition, the double integral of f over R is defined as:
,
provided the limit exists and is the same for all choices of the evaluation points (ui, vi) R for
i = 1, 2, …, n. In this case, we say f is integrable over R.
Theorem 1.1: Fubini’s Theorem (Order of Integration is Interchangeable)
If a function f(x, y) is integrable on the rectangle R = {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d}, then we can write the double integral of f over R as either of the iterated integrals:
.
Definition 1.3: Double Integral of a Function of Two Variables Over Any Bounded Region
For any function f(x, y) defined on a bounded region R, we define the double integral of f over R as:
,
provided the limit exists and is the same for all choices of the evaluation points (ui, vi) Ri for
i = 1, 2, …, n. In this case, we say f is integrable over R.
Theorem 1.2: Double Integral Over a Region with Nonconstant Bounds in the x Direction
If a function f(x, y) is continuous on a bounded region R defined by
R = {(x, y) | a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)} for continuous functions g1 and g2 where g1(x) ≤ g2(x) for all x [a, b], then:
.
Theorem 1.3: Double Integral Over a Region with Nonconstant Bounds in the y Direction
If a function f(x, y) is continuous on a bounded region R defined by
R = {(x, y) | h1(y) ≤ x ≤ h2(y)} c ≤ y ≤ d} for continuous functions h1 and h2 where h1(x) ≤ h2(x) for all y [c, d], then:
.
Theorem 1.4: Linear Combinations of Double Integrals
Let the function f(x, y) and g(x, y) be integrable over the region RR, and let c be any constant. Then the following hold:
- if R = R1R2, where R1 and R2 are nonoverlapping regions, then .
Section 13.2: Area, Volume, and Center of Mass
Double Riemann Sum Over a Region to Compute Volume under the Surface z = f(x, y):
Note: if we pick the upper surface of z = 1, then
Application of Double Integrals: First Moment and Center of Mass
Given a flat plate (or lamina) in the shape of some bounded region R with an area density (x, y), the mass of the lamina is given by:
.
The (first) moments with respect to the x- and y-axes are defined by:
The center of mass of the lamina is given by:
Application of Double Integrals: Second Moment and Moments of Inertia
The (second) moments, or moments of inertia, of a lamina with respect to the x- and y-axes are defined by:
Section 13.3: Double Integrals in Polar Coordinates
Cartesian polar conversion formulas:
x = rcos() and y = rsin(), which implies x2 + y2 = r2
In addition to making the substitution f(x, y) = f(rcos(), rsin()),we also need to:
- Describe the region of integration R in terms of r and , which is usually pretty easy to do.
- Write the differential element of surface area dA in terms of r and , which is just a formula derived below that you have to remember.
A = rrdA = rdrd.
Theorem 3.1: Fubini’s Theorem
If f(r, ) is continuous on the region R = {(r, ) | ≤ ≤ , g1() ≤ r ≤ g2()}, where
0 ≤ g1() ≤ g2() for all [, ], then:
.
Theorem Not-Important-Enough-To-Get-A-Number-Because-No-One-Ever-Uses-It:
If f(r, ) is continuous on the region R = {(r, ) | h1(r) ≤ ≤ h2(r)}, a ≤ r ≤ b}, where
h1(r) ≤ h2(r) for all r [a, b], then:
.
Section 13.4: Surface Area
.
Section 13.5: Triple Integrals
Definition 5.1: The Definite Integral of a Function of Three Variables
For any function f(x, y, z) defined in the rectangular box Q = {(x, y, z) | a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s}, and ||P|| (the norm of the partition) defined as the maximum diagonal of any rectangular box region in the partition, the triple integral off over Q is defined as:
,
provided the limit exists and is the same for all choices of the evaluation points (ui,vi, wi) Qi for
i = 1, 2, …, n. In this case, we say f is integrable over Q.
Theorem 5.1: Fubini’s Theorem (Order of Integration is Interchangeable)
If a function f(x, y) is integrable on the boxQ ={(x, y, z) | a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s }, then we can write the double integral of f over Q as:
.
Definition 5.2: Triple Integral of a Function of Three Variables Over Any Bounded Volume
For any function f(x, y, z) defined on the bounded solid Q, we define the triple integral of f over Q as:
,
provided the limit exists and is the same for all choices of the evaluation points (ui, vi, wi) Qifor
i = 1, 2, …, n. In this case, we say f is integrable over Q.
Special Case of a Triple Integral over a Bounded Volume:
If Q has the form
Q = {(x, y, z) | (x, y) R (a bounded region in the xy plane) and g1(x, y) ≤ z ≤ g2(x, y)}, then
Application of Triple Integrals: First Moment and Center of Mass
Given a shape bounding a region Q with a density (x, y, z), the mass of the shape is given by:
.
The first moment with respect to the yz plane is defined by:
The first moment with respect to the xz plane is defined by:
The first moment with respect to the xy plane is defined by:
The center of mass of the shape is given by:
Section 13.6: Cylindrical Coordinates
Special Case of a Triple Integral over a Bounded Volume Using Cylindrical Coordinates:
If Q has the form
Q = {(r, , z) | (r, ) R and k1(r, ) ≤ z ≤ k2(r, )}, and R has the form
R = {(r, ) | ≤ ≤ and g1() ≤ r ≤ g2() },then
Section 13.7: Spherical Coordinates
Triple Integral over a Bounded Volume Using Spherical Coordinates:
Note:
Section 13.8: Change of Variables in Multiple Integrals
Definition 8.1: Jacobian of a Transformation
The determinant
is called the Jacobian of a transformation T and is written using the notation
Theorem 8.1: Change of Variables in Double Integrals
If a region S in the u-v plane is mapped onto the region R in the x-y plane by the one-to-one transformation T defined by and , where g and h have continuous first derivatives on S, and if f is continuous on R and the Jacobian is nonzero on S, then
.
Change of variables for triple integrals:
In three dimensions, a change of variables is fairly analogous to the two dimensional case:
Given a transformation T from a region S of u-v-w space onto a region R of x-y-z space, specified by the functions
, , and ,