Vector Functions and Space Curves
We now study functions whose values are vectors because such functions are needed
to describe curves and surfaces in space.
Definition 1 (page 848). A vector-valued function, or
Arc Length and Curvature (page 861)
Question. How do we know that two space curves are the same (congruent)?
Concept of a Curve
Definition 1 (page 861). Suppose that C is a space curve given by a vector function
r(t) = f (t) i + g(t) j + h(t) k,
WIN8 64bit QT Creator 5.5.1
Part 1: (40%)
Design a computer program to perform Fourier transform of an image using FFT.
You may use FFT function source code available on the CEIBA course website. The
BASIC DIFFERENTIATION RULES FOR ELEMENTARY FUNCTIONS
sin u cos uu
cot u csc2 uu
Increasing and Decreasing Functions and the First Derivative Test
In Exercises 1 and 2, use the graph of f to find (a) the largest
open interval on which f is increasing, and (b) the la
Applications of Differentiation
(a) Verify that C3 C6.
In Exercises 14, explain why Rolles Theorem does not apply to
the function even though there exist a and b such that f a f b
In Exercises 1 4, complete two iterations of Newtons Method
for the function using the given initial guess.
1. f x x 2 5, x1 2.2
3. f x tan x, x1 0.1
2. f x x3 3
Riemann Sums and Definite Integrals
THEOREM 4.8 PRESERVATION OF INEQUALITY
1. If f is integrable and nonnegative on the closed interval a, b, then
f x dx.
2. If f and g are integrable on the c
Before you begin the exercise set, be sure you realize that one of the most
important steps in integration is rewriting the integrand in a form that fits the basic
The Fundamental Theorem of Calculus
Graphical Reasoning In Exercises 1 4, use a graphing utility
to graph the integrand. Use the graph to determine whether the
definite integral is posit
Concavity and the Second Derivative Test
In Exercises 1 and 2, the graph of f is shown. State the signs of
f and f on the interval 0, 2.
41. f x sin x 1 sin 3x 1 sin 5x, 0,
Triple Integrals, page 1029
Goal: Define and compute triple integrals of f (x, y, z) over a bounded region.
We first deal with the case where f (x, y, z) is defined on a rectangular box:
B = cfw_(x, y, z)|a x b, c y d, r z s.
(1) Divide the box B int
Calculus A (1) 105 Finalterm ANSWER
Problem 1 (12%). Evaluate the integrals.
sin x cos2 x
tan1 ( x) dx.
(a) We use the integration by part to get
dx = csc x sec x dx = csc x d tan x
sin x cos2 x
= csc x tan x tan
Double Integrals over General Regions, page
Goal: We will learn how to integrate a function f (x, y) over a bounded region D.
Define a new function F (x, y) with a rectangular region R D by
f (x, y) if (x, y) is in D
F (x, y) =
if (x, y) is
Double Integrals over Rectangles, page 988
Review of the Definite Integral, page 988
Suppose that f (x) is defined for a x b.
(1) Divide the interval [a, b] into n subintervals [xi1 , xi ] of equal width x =
Calculus A (2) Quiz 2, TEMPLATE
16 True-False (T)-(F).
( ) 1.
( ) 2.
= ln(1 + 2) = ln 3.
( ) 3. If
an 4n is convergent, then
an (2)n is convergent.
Limits and Continuity, page 903
Definition 1 (page 904). Let f be a function of two variables whose domain D
includes points arbitrary close to (a, b). Then we say that the limit of f (x, y) as
(x, y) approaches (a, b) is L ( f (x, y) (a, b) L) and w
Triple Integrals in Cylindrical Coordinates,
Goal: Compute triple integrals in cylindrical coordinates.
Cylindrical Coordinates, page 1040
In the cylindrical coordinate system (), a point P in three-dimensional space
is represented by the o
Tangent Planes and Linear Approximations,
Tangent Planes, page 928
Definition 1 (page 928). Suppose that a surface S has equation z = f (x, y), where
f has continuous partial derivatives, and let P (x0 , y0 , z0 ) be a point on S. Let C1
Maximum and Minimum Values, page 959
Definition 1 (page 960). A function of two variables has a local maximum (
) at (x0 , y0 ) if f (x, y) f (x0 , y0 ) when (x, y) is near (x0 , y0). (This means
that f (x, y) f (x0 , y0 ) for all points (x, y) in so
Change Variables in Multiple Integrals,
Goal: Find relations of change of variable in double and triple integrals.
(1) For a function of one variable f (x), we have the Substitution Rule:
f (x) dx =
f (x(u)x (u) du,
Directional Derivatives and the Gradient Vector, page 946
Directional Derivatives, page 946
Definition 1 (page 947). The directional derivative () of f (x, y) at (x0 , y0 )
in the direction of a unit vector u = (a, b) is
f (x0 + ha, y0 + hb) f (x0 ,
Applications of Taylor Polynomials
In this section we explore some applications of Taylor polynomials. Computer scientists like them because polynomials are the simplest of functions. Physicists and
engineers use them in such fields as re
Calculus A (2) Midterm ANSWER
Problem 1 (10%). Let p (0, 1). A sequence cfw_xn
n=1 is given by
p and xn+1 =
p + xn
for n 1.
Determine whether the sequence is convergent or divergent with an argument. If it
is convergent, find the limit.
Partial Derivative, page 911
Definition 1 (page 913). If f is a function of two variables x and y, suppose we
let only x vary while keeping y fixed, say y = y0 , then g(x) = f (x, y0 ) is a function
of a single variable x. If g(x) has a derivative at
Integration by Substitution
In Exercises 1 4, complete the table by identifying u and du for
f gxg x dx
8x 2 1216x dx
Integration Techniques, LHpitals Rule, and Improper Integrals
Slope Fields In Exercises 33 and 34, a differential equation, a
point, and a slope field are given. (a) Sketch two a
1. (a) The magnitude of r is
| r | (5.0 m) 2 ( 3.0 m) 2 (2.0 m) 2 6.2 m.
(b) A sketch is shown. The coordinate values are in
2. (a) The position vector, according to Eq. 4-1, is r = ( 5.0 m) + (9.0 m)j .
(b) The magnitude is
Double Integrals in Polar Coordinates, page
Goal: We want to evaluate a double integral R f (x, y) dA, where R is easily
described using polar coordinates.
Recall that relations between Cartesian coordinates and polar coordinates:
x = r c
The Chain Rule, page 937
The Chain Rule, Case 1 (page 938). Suppose that z = z(x, y) is a differentiable
function of x and y, where x = x(t) and y = y(t) are both differentiable function of
t. Then z is a differentiable function of t and
z dx z dy
Lagrange Multipliers, page 971
Method of Lagrange Multipliers (page 972). 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 6= 0 on the surface g(x, y, z) =
Derivatives and Integrals of Vector Functions
Derivatives, page 855
Definition 1 (page 855). The derivative () r (t) of a vector function r(t) is
r(t + h) r(t)
= r (t) = lim
Definition 2 (page 856).
(a) The vector r (t0 ) is c
Triple Integrals in Spherical Coordinates,
Goal: Define and compute triple integrals in spherical coordinates.
Spherical Coordinates, page 1045
The spherical coordinates (, , ) () of a point P in space are shown in Figure 1, where = |OP | i