05 2016 - 2017 H2 Maths Differentiation and its Applications Lecture Questions (student) Final

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 1 of 9 National Junior College 2016 – 2017 H2 Mathematics Differentiation and its Applications Lecture Questions Part 1. Key Questions to answer:  How do you differentiate the following functions?  polynomial functions  trigonometric functions  exponential functions  logarithmic functions  constant multiples, sums and differences of any combination of the above functions  How do you use implicit differentiation, parametric differentiation and logarithmic differentiation? Prerequisite knowledge: Differentiation by product rule, quotient rule and chain rule. Lecture Readings: Section 1 Question 1.1 Differentiate the following with respect to x and simplify your answers. (a) 2ln ,1 x x    (b)  2log 1 x . Solution: (a)       2 2 2 2 2 2 22 dd ln ln ln(1 )d 1 d 11 (2 )1 1 2 1 11 x xxx x x xxx x x x x x x x             (b)       2 ln 1ddlog 1d d ln 2 1d ln 1ln 2 d 11 ln 2 1 xxxx xx x    

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 2 of 9 Question 1.2 (Implicit Differentiation) The equation of a curve is given by 3 2 34 3 2x x y y   . Find d d y x in terms of x and y. Solution:   3 2 3 2 2 2 2 2 2 2 22 4 3 2 Differentiating implicitly w.r.t. : dd12 6 3 3 dd d3 3 12 6        d d 4 2             d x x y y x yyx xy x y xx yx y x xy x y x xy x y x                    Question 1.3 (Logarithmic Differentiation) Differentiate (a) tan3 x (b) sin xx with respect to x, leaving your answer in terms of x. Solution: (a) Method 1:    tan 2 tand 3 sec ln 3 3d xx xx  Method 2: Let tan3 xy Taking “ln” on both sides, we get  ln tan ln3yx        2 2 2 tan 1d ln 3 secd d ln 3 secd d sec ln 3 3d x y xyx y yxx y xx    (b) Let sin xyx . Taking “ln” on both sides, we get ln (sin )lny x x . Differentiate implicitly w.r.t. x : 1 d 1 (sin ) (ln )cosd d1 sin (ln )cosd y x x xy x x y y x x xxx     sin 1 sin (ln )cos .xx x x xx  

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 3 of 9 Question 1.4 (Differentiating Inverse Trigonometric Functions) The equation of a curve is given by   1cos 2yx  . Find d d y x in terms of x. Solution:   1 2 cos 2 cos 2 dsin 2 d d 2 2 d sin 14 y x y x yy x y xy x        Question 1.5 (Parametric Differentiation) The curve C is defined parametrically by the equations ln , 1x t t y t    , where 0t . Using parametric differentiation, find d d y x in terms of t. Solution: ln , 1 d 1 d1 , 1dd d d1 d d1d1 d x t t y t xy t t t y yt t xtxt tt              Learning Point(s): (a) Implicit differentiation to find d d y x is useful if y cannot be expressed in terms of the independent variable x explicitly. (b) Logarithmic differentiation is useful for expressions of the form ,vu where both u and v are non- constant expressions of x and/or y. (c) Referring to Question 1.4, alternatively, we can convert the parametric equation into the Cartesian form and differentiate implicitly w.r.t. x. (d) Note that 2 2 2 22 2 d d d dd d y y t xx t  . y 2x 214 x 1

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 4 of 9 Part 2. Key Questions to answer:  What are increasing and decreasing functions?  How do you determine the concavity of a curve?  How do you sketch the graph of the derivative function? Prerequisite knowledge: Curve Sketching, Functions Lecture Readings: Section 2 Question 2.1 Write down the largest set of values of x for which the graph of f ( )yx is (i) increasing, (ii) decreasing, (iii) concave upwards, (iv) concave downwards. Solution: (i) (4, )x (ii) ( ,2)x  or (2,4)x (iii) (2, )xa , where a is x-coordinate of the point of inflexion (iv) ( ,2) ( , )xa    2 2 O a y x 4

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 5 of 9 Question 2.2 (Sketching the derivative function) Sketch the graph of f ( )yx  . Solution: Learning Point(s): (a) The stationary point of f ( )yx becomes the x-intercept of f ( )yx  . (b) The point of inflexion of f ( )yx becomes the stationary point of f ( )yx  . (c) The vertical asymptote of f ( )yx remains unchanged in f ( )yx  . (d) The horizontal asymptote of f ( )yx becomes the horizontal asymptote y = 0 in f ( )yx  . (e) If f ( ) 0x  , then the graph of f ( )yx  lies above the x-axis. If f ( ) 0x  , then the graph of f ( )yx  lies below the x-axis. y f ( )yx 2 (4,1) x a 2 O x y O a 2 4

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 6 of 9 Part 3. Key Questions to answer:  How do you find the gradient of the tangent (and normal) of a curve at a point via implicit and parametric differentiation, and hence the equation of the tangent (and normal)?  How do you solve problems that require you to maximise or minimise a variable?  How do you solve problems that involve connected rates of change? Prerequisite knowledge: Curve Sketching, Tangents and Normals, Differentiation Rules Lecture Readings: Sections 3 and 4 Question 3.1 (Tangent/Normal) Refer to the equation of the curve given in Question 1.2. Find the equations of the tangents to the curve that are parallel to the y-axis. Solution: Equation of curve: 3 2 3 4 3 2            x x y y   As shown in Question 1.2, 2 22 d 4 2 .d y x xy x y x   If the tangent is parallel to the y-axis, then 22 0y x y x    . For y = x: 3 3 3 3 3 1 4 3 2 6 2 3 x x x x x       . For y = − x: 3 3 3 3 4 3 2 2 2 1x x x x x       . Thus equations of tangents that are parallel to the y-axis are 3 1 and 1. 3 xx  Learning Points: (a) The gradient of the tangent to the curve f ( )yx is given by d d y x . (b) What should we do if were asked to find the equations of the tangents that are parallel to the x- axis instead? If the tangent is parallel to the x-axis, then 24 2 0x xy .   24 2 0 4 2 0 0 or 2x xy x x y x y x        When 0x , 3 32 2 yy   . When 2 yx , 32 3 3 24 3 222 3 yy y y y               

National Junior College Mathematics Department 2016 (Student Version) Differentiation and its Applications Page 7 of 9 Question 3.2 (Tangent/Normal) Refer to the equation of the curve given in Question 1.4. (i) Find the equation of the normal to the curve at 1t . (ii) Determine if the normal meets the curve again. Solution: (i) Equation of curve: ln , 1x t t y t    , where 0t . As shown in Question 1.4, d d1 yt xt  . Thus at t = 1, d1 d2 y x  and hence, gradient of the normal is −2. Also, at t = 1, 1, 2xy . Equation of the normal is 2 2( 1) 2 4y x y x      . (ii) 1 2( ln ) 4 3 3 2lnt t t t t       From the graphs above, we see that there is only one intersection