Calculus Notes (PDF)
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Khon Kaen University International College
Manida Suksawat
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These notes provided by Khon Kaen University International College cover calculus topics, including differentiation rules (constant, power, constant multiple, sum), integration rules, indefinite integrals, and a brief introduction to definite integrals. Example calculations are included.
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Calculus Manida Suksawat ([email protected]) Learning objectives Students should be able to understand, explain and calculate: 1.Understanding the rules and being able to calculate the differentiation of calculus 2.Understanding the rules and calculating the integration of cal...
Calculus Manida Suksawat ([email protected]) Learning objectives Students should be able to understand, explain and calculate: 1.Understanding the rules and being able to calculate the differentiation of calculus 2.Understanding the rules and calculating the integration of calculus Calculus Calculus branch of mathematics concerned with the calculation of instantaneous rates of change (differential calculus) and the summation of infinitely many small factors to determine some whole (integral calculus) Rules of differentiation 1. The constant rule 2. The power rule 3. The constant multiple rule 4. The sum rule 1. The constant rule q The constant function f (x) = c is a horizontal line for any constant c. ๐ q The derivative (or the slope) ๐ =๐ ๐ ๐ of a constant function is zero. 2. The power rule For any real number n, ๐ ๐ ๐ = ๐๐๐"๐ ๐ ๐ Examples: % If ๐ฆ = ๐ฅ$ this means that %& ๐ฅ $ = 3๐ฅ ' ! % % ( "! If ๐ฆ= ๐ฅ this means that %& ๐ฅ = %& ๐ฅ " = ' ๐ฅ " 3. The constant multiple rule If c is a constant and f (x) is differentiable, then so is c ยทf (x), and ๐ ๐ ๐๐ ๐ =๐ ๐ ๐ ๐ ๐ ๐ ๐ Examples: % % If ๐ ๐ฅ = 3๐ฅ ) then %& 3๐ฅ ) = 3 %& ๐ฅ ) = 3 4๐ฅ $ = 12๐ฅ $ ! "* % "* % "" ( "# * "# If ๐ ๐ฅ = & then %& & = %& โ7๐ฅ = โ7 โ'๐ฅ " = ' ๐ฅ " 4. The sum rule If f (x) and g(x) are differentiable functions, then so is the sum S(x) = f (x) + g(x), and the derivatives are: ๐ ๐ ๐ ๐ ๐บ ๐ = ๐ ๐ +๐ ๐ = ๐ ๐ + ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ Examples: # # # If ๐ ๐ฅ = ๐ฅ !" + 7 then ๐ฅ !" + 7 = ๐ฅ !" + 7 = โ2๐ฅ !% + 0 = โ2๐ฅ !% #$ #$ #$ # # # If ๐ ๐ฅ = 2๐ฅ & โ 3๐ฅ !' then 2๐ฅ & โ 3๐ฅ !' = 2 ๐ฅ& โ 3 ๐ฅ !' #$ #$ #$ = 2 5๐ฅ ( โ 3 โ7๐ฅ !) = 10๐ฅ ( + 21๐ฅ !) Integration 1. The Indefinite Integral If ๐น ๐ฅ is one antiderivative of the continuous function ๐ ๐ฅ , then all such antiderivatives may be characterized by ๐น ๐ฅ + ๐ถ for constant ๐ถ. The family of all antiderivatives of ๐ ๐ฅ is written as ,๐ ๐ฅ ๐๐ฅ = ๐น ๐ฅ + ๐ถ and is called the indefinite integral of ๐ ๐ฅ. The integral is โindefiniteโ because it involves a constant ๐ถ that can take on any value. Integration Rules for integrating functions Integration Examples: Find the following integrals: 1. 6 3๐๐ฅ Use the constant rule with ๐ = 3: 6 3๐๐ฅ = 3๐ฅ + ๐ถ * 2. 6 ๐ฅ *' ๐๐ฅ Use the power rule with ๐ = 17: 6 ๐ฅ *' ๐๐ฅ = ๐ฅ *) + ๐ถ *) Algebraic rules for indefinite integration Integration Example: Find the following indefinite integral: 6 2๐ฅ & + 8๐ฅ % โ 3๐ฅ " + 5 ๐๐ฅ = 2 < ๐ฅ & ๐๐ฅ + 8 6 ๐ฅ % ๐๐ฅ โ 3 6 ๐ฅ " ๐๐ฅ + 6 5 ๐๐ฅ ๐ฅ+ ๐ฅ( ๐ฅ% =2 +8 โ3 + 5๐ฅ + ๐ถ 6 4 3 1 + = ๐ฅ + 2๐ฅ ( โ ๐ฅ % + 5๐ฅ + ๐ถ 3 Integration 2. The Definite Integral If the function ๐ ๐ฅ is continuous on the interval ๐ โค ๐ฅ โค ๐, then " , ๐ ๐ฅ ๐๐ฅ = ๐น ๐ โ ๐น ๐ ! where ๐น ๐ฅ is any antiderivative of ๐ ๐ฅ on ๐ โค ๐ฅ โค ๐. The numbers ๐ and ๐ are called the lower and upper limits of integration, respectively. The process of finding a definite integral is called definite integration. Integration Example: Find the area of the region under the line ๐ฆ = 2๐ฅ + 1 over the interval 1 โค ๐ฅ โค 3. Since ๐ ๐ฅ = 2๐ฅ + 1 satisfies ๐ ๐ฅ โฅ 0 on the interval 1 โค ๐ฅ โค 3, the area is given by the definite integral % ๐ด = 6 2๐ฅ + 1 ๐๐ฅ * Since an antiderivative of ๐ ๐ฅ = 2๐ฅ + 1 is ๐น ๐ฅ = ๐ฅ " + ๐ฅ, it follows that: % % ๐ด = 6 2๐ฅ + 1 ๐๐ฅ = ๐ฅ " + ๐ฅ D * * = 3" + 3 โ 1" + 1 = 10 Thus the area under the line is 10. Thank you