PDF Understanding Basic Calculus - Universitas Brawijaya

Students should understand the statements of the Intermediate Value Theorem (several versions) and the Extreme Value Theorem. More specifically, the formulas for the derivatives of the sine, cosine and tangent functions as well as logarithmic and exponential functions are given.

Exponents

Algebraic Identities and Algebraic Expressions

Solving Linear Equations

FAQ What is the difference between the word "solution" after the question and the word "solution" in the last sentence. The first "solution" is the solution (answer) to the problem (how to solve the problem), while the second "solution" means the solution of the given equation.

Solving Quadratic Equations

The solution of an equation is the set consisting of all the solutions of the equation. Answer When we extend the real number system to the complex number system, (0.4.1) has two complex solutions if b2−4ac < 0.

Remainder Theorem and Factor Theorem

Proof This follows immediately from the remainder theorem because (x−c) is a factor which means that the remainder. 2) Using long division, we get For each of the following expressions, use the factor theorem to find a linear factor (x−c) and thus factor it completely (using integer coefficients).

Solving Linear Inequalities

We write b ≥ a to indicate that b is greater than or equal to and we write a ≤b to indicate that a is less than or equal to b. The solutions are all real numbers xzodatx<−4, that is, all real numbers less than −4.

Lines

Rectangular Coordinate System Given a plane, there is a one-to-one correspondence between points in the plane and ordered pairs of real numbers (see construction below). Terminology If a line is represented by an equation of the form (0.7.3), we say that the equation is a general linear form for `.

Pythagoras Theorem, Distance Formula and Circles

Solution Using the square of completion method, the given equation can be written in the form (0.8.1). For each of the following, find the distance from the given point to the given line.

Parabola

To rewrite the equation in the form (0.9.1), consider the first two terms and rewrite it in the form x2+b.

Systems of Equations

Introduction

For example, when we say "an old person" (a person is old if they are 65 years old or older), the attribute "old" refers to people. To make this clear, we write. read "the set of all x belonging to R such that x is a positive integer less than 101".

Set Operations

The relative complement of BinA, denoted by A\BorA−B (read "A set minus (or minus) B"), is the set whose elements are those belonging to Abut not belonging to B, i.e. Inside this rectangle, subsets of the universal set are represented by circles, rectangles, or other geometric figures.

Real Numbers

The Number Systems

Real number line Real numbers can be represented by points on a line, called the real number line. So (a,∞) is the set whose elements are the points between and∞, that is, real numbers larger.

Radicals

4 is the principal square root of 4, which is the positive real number whose square is 4. Answer The rules remain valid for rational exponents provided the base is positive (this is required in the definition of bqp).

Solving Inequalities

Quadratic Inequalities

In the following table, the first two rows give the signs (x+5) and (x−3) for each of these intervals. Solving the inequality x2+2x−15 > 0 means finding all x such that the corresponding points on the hasy-coordinate parabola are greater than 0.

Polynomial Inequalities with degrees ≥ 3

If the degree of the polynomial is 3, the cases are 4; if the level is 4, there are 8 cases. In the table method, if the rate increases by 1, the number of factors and the number of intervals increase.

Domains and Ranges of Functions

In terms of solving the equation 3 belongs to the domain means that the equation 3= x2+1 has solution inR(the domain of f). Alternatively, to see that the range is [2,∞), we can use the graph ofy= x2+2, which is a parabola.

Graphs of Equations

In general, a subset 𝒜 of a plane is symmetric with respect to the line ` if the following condition holds: For every point P belonging to 𝒜 (but not belonging to `), there exists a point Q belonging to 𝒜 such that. 1) segment PQ is perpendicular to `;. In general, a subset 𝒜of a plane is asymmetric with respect to a point C if the following condition holds: For every point P belonging to 𝒜 (but different from C), there is a point Q belonging to 𝒜 such that.

Graphs of Functions

Sometimes the square root function is also denoted by √. The position of the dot indicates that the variable is placed there. In general, two subsets of the plane are said to be asymmetric about a line ` if for every point P belonging to one of the two sets there is a point Q belonging to the other set such that either P=Qbelongs to `or. the line segment PQ is perpendicular to `;.

Compositions of Functions

On average, one additional unit will remain vacant for every $500 rent increase. a) Let represent the number of increments of $500. Note If the range of f is not contained in the domain of g, then we must restrict f to a smaller set such that for every x in this set, f(x) belongs to the domain of g. For each of the following, find f(x) and g(x), where g(x) is of the form xrwithr, 1 such that (g◦ f)(x) is equal to the given expression.

Inverse Functions

Finding the inverse function of f means finding the domain of f−1 as well as a formula for f−1(y). 2 = x (x can be solved for all real numbers). Solution Because the domain of is [0,∞), the function is injective. Although the sine function is not injective, we can make it injective by restricting the domain to [−π2,π2].

More on Solving Equations

By direct substitution, we see that there are 7 solutions to the given equation. 2) By multiplying both sides by x(x−1), which is the LCM of the denominators of the terms appearing in the equation, we get By direct substitution, we see that −3 is a solution, but 1 is not a solution to the given equation. First, we divide the interval [0,1] into a finite number of equal subintervals. i, we consider a rectangular region with base on subinterval and height. the largest region that lies under the curve).

Limits of Sequences

Listing a few terms in the sequence is not a good way to describe a sequence. In Problem 1 in the last section, the sequence obtained can be represented by the formula =4+ 1. The method is to discard the constant term 1 in the numerator and denominator (note that if n is very large, compared to n or 2n, 1 is very small).

Figure 3.4 Remark For simplicity, instead of saying Condition (∗), we will say

Limits of Functions at Infinity

The following rules for limits of functions at infinity correspond to the rules for limits of sequences. To find limits at infinity for rational functions, we can divide the numerator and denominator by an appropriate power ofx. However, we cannot apply rule (5) or (6) because the limit at infinity for one of the functions does not exist.

One-sided Limits

Definition Leta∈Rand let f be a function such that f(x) is defined for x sufficiently close to and greater than a. The condition "f(x) is defined for x sufficiently close to and greater than a" means that there is a positive real numberδsuch that f(x) is defined for allx∈(a,a+δ). The graph of f (for x close to and less than 0) is shown in the following figure.

Two-sided Limits

Using the method in the example above, we can prove the following theorem, which implies that the limit of a polynomial function can be found at any real number by substitution. Using the method in the example above, we can prove the following theorem, which implies that the limit of a rational function on each belonging to its domain can be found by substitution. This is because when xi is close to 1, the numerator is close to 2, while the denominator is close to 0, so the fraction is very large (can be positive or negative).

Continuous Functions

We say that f is continuous on I if it is continuous at everya∈(c,d) and is line continuous atc. We say that f is continuous on I if it is continuous at everya∈(c,d) and is left continuous atd. Explanation In the sentence, the condition "f is a function defined and continuous on an interval I" means that "f is a function, I is an interval, I ⊆dom (f) and f is continuous on I".

Rules for Differentiation

Derivative of identity function The derivative of the identity function is the constant function 1, that is d. Since the function is undefined on the left side of 0, we can only consider the differentiability of the function on (0,∞). Rule for derivation of the square root function, the function f is differentiable on (0,∞) and we have f0(x) = d.

Higher-Order Derivatives

For simplicity, some of the concepts and results are not given in their most general form. Many of the concepts and results are given for functions that are differentiable, doubly differentiable, etc. Remark If f is a function defined on an open interval containing x0, we can consider the continuity and differentiability of f atx0.

Curve Sketching

Increasing and Decreasing Functions
Relative Extrema
Convexity
Curve Sketching

In the above examples, we can determine where the function f increases or decreases by inspection or using the graph of f. Suppose that x0 is a critical number of f. 1) If f0(x) changes from positive to negative as xincreases by x0, then x0 is a local maximizer of f. 2) If f0(x) changes from negative to positive as xincreases by x0, then x0 is a local reducer of f. 3) If f0(x) does not change sign as x increases by x0, then x0 is neither a local maximizer nor local. Since f0 is the slope function, f is strictly convex on (a,b), this means that the slope increases and so in the interval (a,b), the graph of f curves up.

Applied Extremum Problems

Absolute Extrema

Note also that f has a relative maximum at x1 but does not reach its absolute maximum there. Example Find the absolute extreme values of the function f given by f(x)=2x3−18x2+30x on the closed interval [0,3]. Since f is continuous on [0,3], it follows from the extreme value theorem that f reaches its absolute extrema.

Figure 5.13 shows the graph of a function f with domain [a, b]. Note that f attains its absolute minimum at x 2 which belongs to the open interval (a, b) and attains its absolute max-imum at b which is an endpoint

Applied Maxima and Minima

Alternative solution to the rectangle problem Let the length of one side of the rectangle be xcm. To increase the volume of the box, the length of the side of the square to be cut is 3 cm. Thus, the length of the side of the square to be cut is 3 cm, and the maximum volume is V(3)=432 cm3.

Applications to Economics

Explanation Although the average cost function is undefined atq= 0, we may include 0 in the domain of the cost function. Find the dimensions of the rectangle with an area of 100 square units that has the smallest perimeter. In the introduction to Chapter 3, we consider the area of the area below the curvey= x2 and above the x-axis for x between 0 and 1.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus, Version 1 Let f be a function that is continuous on a closed and bounded interval [a,b]. From the above example, we see that given a function f that is continuous on a closed and bounded interval [a,b], if we can find a primitive form over [a,b], then we can find the integral designated Rb. The Fundamental Theorem of Calculus, Version 2 Let f be a function that is continuous on a closed and bounded interval [a,b].

Indefinite Integrals

Proof The result follows from the Constant Multiple Rule and Power Rule (negative integer version) for . Proof The result follows from the Constant Multiple Rule and Power Rule (n+12 version) for Differentiation. Sum rule for integration (Term for term integration) Let fandgbe be functions that are continuous on an open interval (a,b).

Application of Integration

Example Find the area of the combined region bounded by the curve=x3−5x2+6x and the x-axis. For each of the following, find the area of the region bounded by the given curve, the x-axis, and the given vertical lines. For each of the following, find the area of the (combined) region bounded by the given curves (or lines).

Trigonometric Functions

To determine the correct sign, assume that x belongs to the 1st quadrant and therefore (π−x) belongs to the 2nd quadrant. According to the CAST rule, sin(π−x) is positive and cos(π−x) is negative (and sinx and cosx are positive). To determine the correct sign, assume that x belongs to the 1st quadrant and therefore 3π.

Differentiation of Trigonometric Functions

Example For each of the following, find dx. dxcosx Term-by-term differentiation and the constant multiple rule. Let f(x)=sin(ax+b) and letg(x)=cos(ax+b) and equal constants. c) Think of the formula for f(n)(x) and g(n)(x) for the general(positive integer). FAQ Instead of the above sequence (an), can we get other sequences (bn) of rational numbers converging to √.

Logarithmic Functions

Differentiation of Exp and Log Functions

Implicit Differentiation

More Curve Sketching