2.3.7 Derivative of Trigonometric Function: Trigonometric Rules 1. The following rules can be applied. a. ������ = sin(������) b. ������ = cos(������) d. ������������ = cos(������) f. ������������ = − sin(������) ������������ ������������ c. ������ = tan(������) ������ = sec(������) ������������ = sec2(������) ������������ = sec(������) ∙ tan(������) ������������ ������������ e. ������ = csc(������) ������ = cot(������) ������������ ������������ = −csc(������) ∙ cot(������) ������������ = −csc2(������) ������������ Tips: If the function is given as ������ = sin ������, then rewrite the function as ������ = sin(������). 2. Let ������ = ������(������), then the Chain Rule can be used to differentiate ������. a. ������ = sin(������) b. ������ = cos(������) ������������ = cos(������) ∙ ������′ ������������ = − sin(������) ∙ ������′ ������������ ������������ c. ������ = tan(������) d. ������ = sec(������) ������������ = sec2(������) ∙ ������′ ������������ = sec(������) ∙ tan(������) ∙ ������′ ������������ ������������ e. ������ = csc(������) f. ������ = cot(������) ������������ = −csc(������) ∙ cot(������) ∙ ������′ ������������ = −csc2(������) ∙ ������′ ������������ ������������ Example 3.11 Differentiate the following functions. a. ������ = sin(−5������) b. 1 ������ = cos (������2) 43
c. ������ = tan(2 + 3������) d. ������ = ������ ∙ sec ������3 e. ������ = csc(tan(3������)) f. ������ = cot √������ 44
3. The power rule and chain rule can also be applied to differentiate trigonometric function as demonstrated as follows. Example 3.12 Differentiate the following function. ������ = √sin(������ − 3) Example 3.13 Differentiate the following function. ������ = cos4 ������ 45
Example 3.14 Differentiate the following function. 3������ ������(������) = tan4 ������ Example 3.15 Differentiate the following function. ������(������) = cot ( 2 ) √3 − ������ 46
Tutorial 2.4 b. ������ = cos ������−2 Differentiate the following functions. a. ������ = sin √������ + 2 c. ������ = tan(10������2 + ������) d. ������ = sec(5������) e. ������ = csc(−3������) f. 1 ������ = −cot ( ) √������ 47
Differentiate the following functions. h. ������ = √cos(������) + 3 g. ������ = (sin 2������)12 i. ������ = tan3(������−3) j. 2 ������ = sec 5������ k. ������ = csc(1 − sin 3������) l. ������ = cot 3������ ∙ cos2 2������ 48
2.3.8 Derivative of Exponential Function: Exponential Rule Let ������ be a real number with the exponent ������ Let ������ be a mathematical constant with ������ = ������������ exponent ������ ������������ ������ = e������ ������������ = ������������ ∙ ln(������) ������������ = e������ ������������ Let ������ = ������(������), then the Chain rule can be applied to differentiate ������. ������ = ������������ ������ = e������ ������������ = ������������ ∙ ln(������) ∙ ������′ ������������ = e������ ∙ ������′ ������������ ������������ Example 3.16 Differentiate the following functions. a. ������ = 54������ b. ������ = esin ������ c. ������ = ������ ∙ e������ d. ������ = tan(sin(e������)) 49
Tutorial 2.5 Differentiate the following functions. a. ������ = e������ − 1 − ������2 + 3������ b. ������ = ������2 ∙ 2������ e������ c. e������ d. ������ = (e������ + cos ������)3 ������ = ������2 e. ������ = e√3−������ f. ������ = sin(e������) 50
2.3.9 Derivative of Logarithmic Function: Logarithmic Rule ������ = log������(������) ������ = ln(������) ������������ 1 ������������ 1 ������������ = ������ ∙ ln(������) ������������ = ������ Tips: If the function is given as ������ = ln ������, rewrite the function as ������ = ln(������). Let ������ = ������(������), then the Chain rule can be applied to differentiate ������. ������ = log������(������) ������ = ln(������) ������������ = ������ ∙ 1 ∙ ������′ ������������ = 1 ∙ ������′ ������������ ln(������) ������������ ������ Example 3.17 Differentiate the following functions. a. ������ = log3(������) b. ������ = ln(10������2) c. ������3 d. ������ = sec(2 ln ������) ������ = ln ������ 51
Example 3.18 Use the properties of logarithm to differentiate the following function. cos(2������) + 3 ������ = ln ( (1 + ������2)4 ) 52
Tutorial 2.6 Differentiate the following functions. a. 1 b. ������ = ln(2e������ + 3) ������ = ln ������ + ������ c. ������ = ln(sin 3������) d. ������ = (tan(ln ������))2 e. −ln ������ f. ������ = 2 ∙ 3√ln 2������ ������ = 2������ 53
2.3.10 Implicit Differentiation 1. It is also possible to differentiate a function of other variables (for example: ������(������), ������(������), and so on) with respect to ������. Example 3.19 Find ������������ by differentiating the following function implicitly with respect to ������. ������������ ������2 + ������3 = 4 sin ������ 54
Example 3.20 Find ������������ by differentiating the following function implicitly with respect to ������. ������������ 2������������ − ������ ∙ e2������ = ln(tan ������) 55
Example 3.21 Find ������������ by differentiating the following function implicitly with respect to ������. ������������ 1 4 = 3√������2 − cos(3������) ������2 + 56
Tutorial 2.7 Differentiate the following functions implicitly. a. ������2 − 2������ ∙ cos ������ + 3������ = ln ������ 57
Find the derivative of the following function using implicit differentiation. b. sin(������ + ������) = cos(2������) − 3e2������ 58
Find the derivative of the following function using implicit differentiation. c. ln(������������) = 3√������ − tan(������������) 59
Find the derivative of the following function using implicit differentiation. d. 1 − 3������4 + sec ������ = etan ������ ln ������ 60
Chapter 3 Applications of Differentiation 3.1 Tangent Lines 1. Tangent line is a straight line equation that “touches” a curve at a point (������, ������). (������, ������) Tangent line Tangent line of a curve ������ = ������(������) at point (������0, ������0) Step 1: Calculate the slope ������ ������������ ������ = ������������ - Use suitable differentiation technique to differentiate ������(������). - Substitute the value of ������0 and ������0 into ������. Step 2: Find the equation of tangent line using the formula ������ = ������������ + ������ or ������ − ������0 = ������(������ − ������0) Example 3.1 Given a curve ������ = 3������2 + 2 and point ������(1,5). Find the tangent line of the curve at ������. 61
Example 3.2 Find the equation of tangent line of the function at the point (2, −6). 2������2 − 2 ������(������) = 1 − ������ Example 3.3 Given the function 4������e2������ − tan ������ = ln(������2). Find the gradient of the tangent line for the function at (1, 0). 62
Tutorial 3.1 Find the equation of tangent line of the function at the given point. a. ������(������) = (������ − 1)3 − 2������2; (1, −2) b. ������2������ − ������2������ = ������2 + 3; (−3, −4) 63
3.2 Linear Approximation and Differentials Approximate a value of a quantity using differentials ������������ Step 1: Identify ������0, ������������ and ������(������). Step 2: Find ������′(������) and ������′(������0). Step 3: Use ������(������0 + ������������) ≈ ������(������0) + (������′(������0) ∙ ������������). Example 3.4 Use differentials to estimate the value of the given quantity. Give answer correct to four decimal places. √81.025 64
Example 3.5 Estimate the value correct to four decimal places using differentials. 2 √63.99 − 3√63.99 Example 3.6 Use differentials to approximate the following value, correct to three decimal places. 3 (4.02)2 + (4.02)2 65
Tutorial 3.2 Find the vertical and horizontal asymptotes of the following functions. a. 5.022 − 2 (5.02 − 2)3 b. cos 89° 66
3.3 Related Rates 1. The problem involves finding the rate of change of a quantity related to other quantities with respect to time ������. Solving problems with related rates Step 1: Identify the rate of change that is given and unknown (to calculate). Step 2: Construct an equation that relates the variables in Step 1. Step 3: Differentiate the equation in Step 2 implicitly with respect to time ������. Step 4: Substitute the given values in Step 1 into equation in Step 3. Step 5: Solve the unknown rate of change. Example 3.7 A spherical ball expands at the rate of 6 cm3/s. Find the rate of change of its radius when the radius is 3.5 cm. 67
Example 3.8 Water is being poured into a cylindrical can at a rate of 4 cm3/s. The can has a radius of 2 cm and height 12 cm. How fast is the height changing at an instant when the height is 5 cm? 68
Tutorial 3.3 Find the related rates for the given problems. a. The surface area of a sphere ������ cm2 is given by the formula ������ = 4������������2, where ������ is the radius in cm. Given that ������ is increasing at a constant rate of 0.2 cm/s, find the rate of change of ������ at the instant when ������ = 2.5 cm. b. The area of a triangle is decreasing at rate of 5 cm2/min. Find the rate of change of the side, when the area of the triangle is 220 cm2. 69
3.4 Graph of Polynomial Function 1. The concepts of differentiation can be used to sketch polynomial functions. First Derivative Test: Find the intervals where a function ������(������) is increasing/decreasing. Step 1: Find ������′(������). Step 2: Find the critical points when ������′(������) = 0. Step 3: Build the sign table for ������′(������) with intervals using the critical points in Step 2. Step 4: Test increase/decrease a. If ������′(������) > 0, then ������(������) increase at the interval. b. If ������′(������) < 0, then ������(������) decrease at the interval. Second Derivative Test: Find the intervals where ������(������) is concave up/concave down. Step 1: Find ������′′(������). Step 2: Find the critical points when ������′′(������) = 0. Step 3: Build the sign table for ������′′(������) with intervals using the critical points in Step 2. Step 4: Test concavity a. If ������′′(������) > 0, then ������(������) is concave up at the interval. b. If ������′′(������) < 0, then ������(������)is concave down at the interval. Extremum Point: The coordinate (������, ������) of the critical point where ������(������) changes from a. Increase to Decrease b. Decrease to Increase Inflection Point: The coordinate (������, ������) of the critical point where ������(������) changes from a. Concave Up to Concave Down b. Concave Down to Concave Up 70
Example 3.9 Given the function ������(������) = ������3 − 6������2 + 9������ + 2 a. Find ������ −intercept(s) of ������(������). b. Find the interval(s) where ������(������) is increasing or decreasing. Hence, determine the extremum point(s), if any. 71
c. Find the interval(s) where ������(������) is concave up or concave down. Hence, determine the inflection point(s), if any. d. Sketch the graph of ������(������) using the above information. 72
Example 3.10 Given the function ������(������) = ������3 − 12������ a. Find the interval(s) where ������(������) is increasing or decreasing. b. Hence, determine the extremum point(s), if any. 73
c. Find the interval(s) where ������(������) is concave up or concave down. d. Hence, determine the inflection point(s), if any. e. Sketch the graph of ������(������) using the above information. 74
Tutorial 3.4 a. Given the function ������ = (������ − 1)(������2 − 1) i. Find the ������ −intercept(s) and ������ −intercept(s). ii. Find the interval(s) where ������(������) is increasing or decreasing. Hence, determine the extremum point(s), if any. 75
iii. Find the interval(s) where ������(������) is concave up or concave down. Hence, determine the inflection point(s), if any. iv. Sketch the graph of ������(������) using the above information. 76
b. Given the function ������(������) = ������4 − 2������3 − 3������2 i. Find the ������ −intercept(s) and ������ −intercept(s). ii. Find the interval(s) where ������(������) is increasing or decreasing. Hence, determine the extremum point(s), if any. 77
iii. Find the interval(s) where ������(������) is concave up or concave down. Hence, determine the inflection point(s), if any. iv. Sketch the graph of ������(������) using the above information. 78
3.5 Graph of Rational Function 1. The concept of limits is used to find the vertical and horizontal asymptotes for rational functions. Example 3.11 Given the function ������(������) = ������ + 3 ������ − 2 a. Find ������ −intercept and ������ −intercept. b. Find the vertical and horizontal asymptotes. 79
c. Find the interval(s) where ������(������) is increasing or decreasing. Hence, determine the extremum point(s), if any. d. Find the interval(s) where ������(������) is concave up or concave down. Hence, determine the inflection point(s), if any. e. Sketch the graph of ������(������) using the above information. 80
Example 3.12 Given the function ������(������) = 2 − 5������ ������ − 3 a. Find ������ −intercept and ������ −intercept. b. Find the vertical and horizontal asymptotes. 81
c. Find the interval(s) where ������(������) is increasing or decreasing. Hence, determine the extremum point(s), if any. d. Find the interval(s) where ������(������) is concave up or concave down. Hence, determine the inflection point(s), if any. e. Sketch the graph of ������(������) using the above information. 82
Tutorial 3.5 a. Given the function 3������ ������ = ������ + 4 i. Find the ������ −intercept and ������ −intercept. ii. Find the vertical and horizontal asymptotes. 83
iii. Find the interval(s) where ������(������) is increasing or decreasing. Hence, determine the extremum point(s), if any. iv. Find the interval(s) where ������(������) is concave up or concave down. Hence, determine the inflection point(s), if any. v. Sketch the graph of ������(������) using the above information. 84
b. Given the function ������(������) = 1 − 2������ 2 + ������ i. Find the ������ −intercept and ������ −intercept. ii. Find the vertical and horizontal asymptotes. 85
iii. Find the interval(s) where ������(������) is increasing or decreasing. Hence, determine the extremum point(s), if any. iv. Find the interval(s) where ������(������) is concave up or concave down. Hence, determine the inflection point(s), if any. v. Sketch the graph of ������(������) using the above information. 86
3.6 Applied Maximum and Minimum Problems 1. The first and second derivative tests can be used in the application of maximum or minimum problem. Maximum or Minimum Problem Step 1: Draw and label a figure with suitable variables that represents the problem. Step 2: Find an equation of the quantity to be maximized or minimized Step 3: Find the fixed quantity that is given in the problem. Step 4: Write the equation in terms of one variable. Step 5: Use the first derivative to find the value of the quantity. Step 6: Use the second derivative test to check if the quantity is maximum or minimum. Example 3.13 Past Semester Question: January 2018 Q2 A piece of wire of length 130 cm is bent into the shape ������������������������������������������. Given ������������ = ������������. ������ ������ 4������ cm ������ ������ ������ (������ + 5) cm ������ ������ cm ������ a. Express ������ in terms of ������. 87
b. Show that the area enclosed by the wire is ������ = 200������ − 12������2. c. Hence, find the maximum area of the shape ������������������������������������������. 88
Example 3.14 A rectangular piece of cardboard of 18 cm wide and of 30 cm length is used to construct an open box. A square of sides x cm is cut from each of the corner and then the sides are bending up. a. Show that the volume of the box is 540������ − 96������2 + 4������3. b. Find the dimension of the corner square for which the volume is maximized. 89
Example 3.15 A prism has right triangular base with sides 3������ cm, 4 cm, and 5 cm. The height of the prism is ������ cm. Given the volume of the solid is 390 cm3. a. Show that the total surface area is ������ = 12������2 + 780 ������ b. Determine the values of ������ and ������ so that the surface area of the prism is minimum. 90
Tutorial 3.6 a. A closed rectangular box has the length twice its width. Given the total surface are of the box is 300 cm2. If the width of the box is ������ cm and the volume of the box is ������ cm3. i. Show that the volume of the box is given by ������ = 100������ − 4 ������3 3 ii. Find all the dimensions of the box when the box is maximized. 91
b. Past Semester Question: March 2017 Q3 The volume of a cylindrical can open at the top with height, ℎ cm and radius, ������ cm is 1200 cm3. Find the minimum surface area of the cylindrical can. 92
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