Calculus Notes (PDF)

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. 𝒅 q The derivative (or the slope) 𝒄 =𝟎 𝒅𝒙 of a constant function is zero. 2. The power rule For any real number n, 𝒅 𝒏 𝒙 = 𝒏𝒙𝒏"𝟏 𝒅𝒙 Examples: % If 𝑦 = 𝑥$ this means that %& 𝑥 $ = 3𝑥 ' ! % % ( "! If 𝑦= 𝑥 this means that %& 𝑥 = %& 𝑥 " = ' 𝑥 " 3. The constant multiple rule If c is a constant and f (x) is differentiable, then so is c ·f (x), and 𝒅 𝒅 𝒄𝒇 𝒙 =𝒄 𝒇 𝒙 𝒅𝒙 𝒅𝒙 Examples: % % If 𝑓 𝑥 = 3𝑥 ) then %& 3𝑥 ) = 3 %& 𝑥 ) = 3 4𝑥 $ = 12𝑥 $ ! "* % "* % "" ( "# * "# If 𝑓 𝑥 = & then %& & = %& −7𝑥 = −7 −'𝑥 " = ' 𝑥 " 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: 𝒅 𝒅 𝒅 𝒅 𝑺 𝒙 = 𝒇 𝒙 +𝒈 𝒙 = 𝒇 𝒙 + 𝒈 𝒙 𝒅𝒙 𝒅𝒙 𝒅𝒙 𝒅𝒙 Examples: # # # If 𝑓 𝑥 = 𝑥 !" + 7 then 𝑥 !" + 7 = 𝑥 !" + 7 = −2𝑥 !% + 0 = −2𝑥 !% #$ #$ #$ # # # If 𝑓 𝑥 = 2𝑥 & − 3𝑥 !' then 2𝑥 & − 3𝑥 !' = 2 𝑥& − 3 𝑥 !' #$ #$ #$ = 2 5𝑥 ( − 3 −7𝑥 !) = 10𝑥 ( + 21𝑥 !) Integration 1. The Indefinite Integral If 𝐹 𝑥 is one antiderivative of the continuous function 𝑓 𝑥 , then all such antiderivatives may be characterized by 𝐹 𝑥 + 𝐶 for constant 𝐶. The family of all antiderivatives of 𝑓 𝑥 is written as ,𝑓 𝑥 𝑑𝑥 = 𝐹 𝑥 + 𝐶 and is called the indefinite integral of 𝑓 𝑥. The integral is “indefinite” because it involves a constant 𝐶 that can take on any value.  Integration Rules for integrating functions  Integration Examples: Find the following integrals: 1. 6 3𝑑𝑥 Use the constant rule with 𝑘 = 3: 6 3𝑑𝑥 = 3𝑥 + 𝐶 * 2. 6 𝑥 *' 𝑑𝑥 Use the power rule with 𝑛 = 17: 6 𝑥 *' 𝑑𝑥 = 𝑥 *) + 𝐶 *) Algebraic rules for indefinite integration Integration Example: Find the following indefinite integral: 6 2𝑥 & + 8𝑥 % − 3𝑥 " + 5 𝑑𝑥 = 2 < 𝑥 & 𝑑𝑥 + 8 6 𝑥 % 𝑑𝑥 − 3 6 𝑥 " 𝑑𝑥 + 6 5 𝑑𝑥 𝑥+ 𝑥( 𝑥% =2 +8 −3 + 5𝑥 + 𝐶 6 4 3 1 + = 𝑥 + 2𝑥 ( − 𝑥 % + 5𝑥 + 𝐶 3  Integration 2. The Definite Integral If the function 𝑓 𝑥 is continuous on the interval 𝑎 ≤ 𝑥 ≤ 𝑏, then " , 𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹 𝑎 ! where 𝐹 𝑥 is any antiderivative of 𝑓 𝑥 on 𝑎 ≤ 𝑥 ≤ 𝑏. The numbers 𝑎 and 𝑏 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 𝑦 = 2𝑥 + 1 over the interval 1 ≤ 𝑥 ≤ 3. Since 𝑓 𝑥 = 2𝑥 + 1 satisfies 𝑓 𝑥 ≥ 0 on the interval 1 ≤ 𝑥 ≤ 3, the area is given by the definite integral % 𝐴 = 6 2𝑥 + 1 𝑑𝑥 * Since an antiderivative of 𝑓 𝑥 = 2𝑥 + 1 is 𝐹 𝑥 = 𝑥 " + 𝑥, it follows that: % % 𝐴 = 6 2𝑥 + 1 𝑑𝑥 = 𝑥 " + 𝑥 D * * = 3" + 3 − 1" + 1 = 10 Thus the area under the line is 10. Thank you