Differential Calculus Average Rate of Change (AROC) The average rate of change of y over an interval is equal to

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2 Differential Calculus Average Rate of Change (AROC) The average rate of change of y over an interval is equal to change in y y y1 f ( x) f ( x1 ) f ( b) f ( a). changein x x x x x b a 1 Example: Find the average rate of change of the function with rule y x x 5 as x changes from 1 to 5. 1 Instantaneous Rate of Change & 1 st Principles If we look at the graph on the right, y x and wanted to calculate the rate of change at Point P, we then calculate the gradient between P and Q. If we bring the point Q closer and closer to P then the gradient will be approaching the value of the tangent at P. ( a h) a a ah h a ah h m( PQ) a h a h a h h If Q approaches P then h 0, the gradient approaches a. The instantaneous rate of change of a function f at point P on a graph of y f (x) is equal to the gradient of the tangent to the graph at P. So, to find the instantaneous rate of change at point P, we evaluate the derivative of the function at P. o The instantaneous rate of change of f at lim f ( x h) f ( x) Example: Find for h 0 h (i) f ( x) 3x x (ii) x a is f '( a). f ( x) x 3 Ex9A 1,, 3, 4 LHS, 8 LHS

3 n The derivative of x If f n ( x) x then n1 f '( x) nx and if f n ( x) ax then n1 f '( x) nax. The derivative of a constant If f ( x) c then f '( x) 0. Examples: Find the derivative of the following: 6 1. y 3x 4x 3. f ( x) 3x(x 7) 3. f g 5 4 6a 7a 4. h a y x 3 x x Remember to always to subtract 1 from the power. Be careful with + and signs. Solutions Ex9B 1,, 4, 5, 6

4 The Gradient of a Curve The gradient of a curve is not constant. The gradient of a curve at a certain point is equal to the gradient of the tangent to the curve at that point. A tangent is a line that touches another curve at one point only (i.e.it does not cross it). a T1 b T The line T1 is a tangent to the curve at point a. The line T is a tangent to the curve at point b. Consider y 3 4x 8x and its derivative 1x 16x What does all this mean? y 3 4x 8x is a formula that gives the y-value of the curve at any point x. 1x 16x is a formula that gives the gradient of the curve at any point x. Example 1: What is the gradient of the curve Solution: y 3 4x 8x at x?

5 Example : What are the co-ordinates of the point(s) of the curve gradient is 4? Solution: y 3 4x 8x, where the Ex 9B 7, 11, 1, 13, 14, 16, 17 Ex 9C 4, 5, 6, 8, 10 Notes: {x: h (x) > 0 } Means: {x: } Find the where h (x) The function, > 0 that is m = tan ɵ opp O tan adj A

6 Strictly increasing and strictly decreasing functions A function f is said to be strictly increasing when a < b implies f(a)<f(b) for all a and b in its domain. The definition does not require f to be differentiable, or to have a non-zero derivative, for all elements of the domain. If a function is strictly increasing, then it is a one-to-one function and has an inverse that is also strictly increasing. If f ( x) 0 for all x in the interval then the function is strictly increasing. If f ( x) 0 for all x in the interval then the function is strictly decreasing. 3 Example 1: The function f : R R, f ( x) x is strictly increasing with zero gradient at the origin The inverse function f : R R, f ( x) x, is also strictly increasing, with a vertical tangent of undefined gradient at the origin. 1 Example : The hybrid function g with domain [0, ) and rule: x 0 x g ( x) is strictly increasing, and is not x x differentiable at x =. Example 3: Consider h : R R, h( x) x x H is not strictly increasing, But is strictly increasing over the 1 interval 0,. 3 3

7 Strictly Decreasing A function f is said to be strictly decreasing when a < b implies f ( a) f ( b) for all a and b in its domain. A function is said to be strictly decreasing over an interval when a < b implies f ( a) f ( b) for all a and b in its interval. Example 4: The function f : R R, f ( x) x e 1 1 The function is strictly decreasing over R. Example 5: The function g : R R, g( x) cos( x) g is not strictly decreasing. But g is strictly decreasing over the interval 0,. Also [, ], [, ], etc. Ex9B 18, 19, 0, 1,

8 Sketching the Gradient Function GRAPH OF THE ORIGINAL FUNCTION Where the gradient is flat (i.e at all stationary points) Where there is a positive gradient ( i.e. slopes ) Where there is a negative gradient ( i.e. slopes ) Where the gradient gets flatter Where the gradient gets steeper At the steepest part of each section of the graph GRAPH OF THE GRADIENT FUNCTION Will cross the x axis Will be above the x axis Will be below the x axis Gets closer to x axis Gets further away from x axis Will have a peak Example 1: Sketching the Gradient Function y = 4x x 4 5 y x 1 / x

9 Example : Sketch the gradient graph of: 4 y x - -4 Solution:

10 Example 3: Sketch the gradient function of: y x - -4 Solution: Ex9D 1 acdefhi, acdegi, 3, 5, 6, 7

11 Chain Rule The derivative of (function) n (The function in a function rule or Composite Function rule). Example 1: Find the derivative of y 4 3(14x x). Solution: In words: find the derivative of the thing as a whole, then multiply it by the derivative of the inside. In symbols: If y ) n ( u where u f (x), then du du d f ( g( x)) g' ( x). f '( g( x)) Let du u 14 x x, 4 y 3u du 3 8x 1, 1u 1(14x x 1(14x Example : Find f '( x) if 3 x) (8x 1) etc 4 18 f ( x) (x 3). ) 3 Example 4 d d f ( x ) f ( x) Example 5 3 Example 3: If y x 3 3 then find. Ex9E 1,, 3, 5

12 Differentiating Rational Powers Examples: Find the derivative of each of the following with respect to x. 7 (a) y 3x (b) 5 x f 3 ( x) x x Ex9F, 3,4, 6, 7

13 Derivatives of Transcendental functions kx The derivative of e kx kx In general: If y e then ke. If (x y ae f ) then Example 1: Find the derivatives of: x (i) y e (ii) (iii) y e y e 5x Solutions: (i) ( x x) af '( x) e f ( x) (ii) (iii) Example : Find f '( x) given f 4x ( x) x e. Solution: Ex9G 1,, 3, 4, 5, 6

14 Derivative of log e x 1 In general, if y log e x then x 1 h'( x) If y log e( h( x)) then h'( x) h( x) h( x) y log x e log e x, x 0 If y log e( x), x 0 1, x 0 x 1, for x R\{0} 1 1 1, x 0 x x x Examples: Find the derivatives of: (i) y log e 3x (ii) y log e( x x) (iii) y log e x x Solution: (i) (ii) (iii) Ex9H 1,, 3, 4, 5, 6, 7, 8

15 Derivative of the Trigonometric Functions If y sin(kx) then k cos(kx) If y cos(kx) then k sin(kx) k or cos ( kx) y tan(kx) then k sec ( kx) Examples: Find the derivative of the following: x 1 (i) y cos cos x 3 3 (ii) y sin( x 3 ) 3 (iii) y sin x (sin x) (iv) y 3tan( x) (v) y cos(3 x ) Solution: (i) (ii) (iii) 3 (iv) (v) Ex9I 1,, 3, 4, 5, 6 NOTE: Angle MUST be in RADIANS c o 180 e. g. sin d x o x sin 180 o d x x o sin x sin cos or cosx

16 The Product Rule The derivative of the product of two functions Example 1: Find (using the product rule) if 3 ( y x x x ). Solution: In words: The derivative of the first term multiplied by the second term, ADD the derivative of the second term multiplied by the first term. du dv In symbols: If y u. v then v. u. Example : Find f '( x) if f ( x) x(4x 3 1). Solution: x Example 3: Find f '( x) if f ( x) e sin( x 1). Solution: Ex9J 1,, 3, 4, 5, 6, 7, 8

17 Quotient Rule The derivative of the quotient of two functions Used when you have a problem in fraction form. Example 1: If y x 4 3 x 7 then find Solution: In words: The derivative of the top term, multiplied by the bottom term, subtract the derivative of the bottom term, multiplied by the top term, all over the bottom term squared. In symbols: If du dv v. u. u y then, v v Example : If Solution: x y x 1 then find. 1 x e Example 3: Find if y. x e 1 Solution: Ex9K 1 aceg, de, 4, 5a, 6a, 7

18 Continuous functions and Differentiable Functions The graph of a continuous function is one without breaks. It is usually a smooth unbroken curve, however it may have sharp corners. If the derivative of a function exists at a point on a curve this function is said to be differentiable at this point. The derivative exists at a point if it is possible to draw a tangent at that point. i.e. the curve must be smooth and continuous. y y 10 5 x x x= At x 1, the graph is continuous and differentiable. At x 1, the graph is continuous BUT NOT differentiable. y x At x 1, the graph is neither continuous or differentiable. -4 Ex9L, 3, 4 Ex9M 1,, 3, 5 Note: No derivative exists at: CUSP point END-POINT (open or closed) HOLE point

19 Finding the Equation of a Tangent and a Normal y=f(x) y P(x,y) P ( x, y) is a point on the curve y f (x). The Normal and Tangent are at right angles to each other. Normal at P If m = gradient of the curve at P, then the gradient of the tangent at Point P = m The gradient of the normal at point P is Tangent at P 1 m x The equation of the Tangent is: y y1 m( x x1). The equation of the Normal is: 1 y y1 ( x x 1 ). m How would you find m if you knew the equation of the curve? Find and substitute the x - coordinate of P into it. Example 1: Find the equation of the tangent and of the normal to the curve point ( 0,1). y ( x 1) 9 at the Ex10A 1,, 3, 6, 8ac, 9abc, 14, 16

20 Rates of Change What is a rate? If you work and earn $1 an hour your rate of pay = $1 per hour = $1/hr. This is linked with calculus by If P = total Pay ($) & t = time worked (hr) The P 1t dp & 1 dt dp rate of change of P with respect to t. dt dp $ For the unit of, $ per hour,. dt hr If you have to find The rate of change of volume with respect to the radius Choose letters for the variables The rate you need is So you ll need an equation relating Unit of rate is The rate of increase of cost of production of dolls w.r.t the number of dolls The rate of change of circumference w.r.t height The rate of decrease of amount of water in a draining tank In the last case, what s missing? w.r.t nd variable assumes it is time. Solving a rate problem is very, very similar to solving max/min prob. 1. need what rate? (no second variable assume time). find a formula. 3. formula must be in terms of one variable only, if not a relationship between the variables by other info. From question. 4. find the rate. 5. substitute given value of second variable, include units 6. answer all questions. If rate is positive, it is increasing, if the rate is negative, it is decreasing.

21 Example 1: A spherical balloon is being inflated. Find the rate of increase of volume with respect to the radius when the radius is 10cm. Solution: Example : The amount of water in a tank (A litres) at any time (seconds) is given by the rate of change of A when Solution: t 5s. A 3. Find t

22 3 Example 3: A balloon develops a microscopic leak. It s volume V ( cm ) at time, t (s) is: t V t, t (i) At what rate is the volume changing when t = 10 seconds? (ii) What is the average rate of change of volume in the first 10 seconds? (iii) What is the average rate of change of volume in the time interval from t 10 to t 0 seconds? Solution: Part (i) above is an INSTANTANEOUS rate of change, Part (ii) & (iii) is an AVERAGE rate of change, i.e. and average of a number of instantaneous rates.

23 Particular Case Displacement Velocity Acceleration Symbol Units Definition Displacement x, x(t), s(t), d m, km, The distance from a fixed point O Velocity Acceleration ds m/s, ms -1, v,, dt dt km/h, dv d x m/s, ms -, a,, dt dt km/h The rate of change of displacement The rate of change of velocity Original displacement/velocity/acceleration occurs at t = 0 NOTE: If you were asked to find the average rate of velocity, it would be done as an average rate of x x1 change ( i.e. ) using the displacement values not the velocity values. (If the velocity values t t1 were used then you get the average acceleration!) Ex10B 1,, 4, 8, 10, 1, 13

24 Finding the Stationary Points of a Curve Example 1: Sketch the graph of f ( x) ( x 1)( x )( x 3) and determine the coordinates of all turning points ( d.p.). Solution: 1.Find x and y intercepts: X-Int ( y = 0) Y-Int ( x = 0). Stationary Points ( 0): 3 Type of Stationary Points: o Local Minimum o Local Maximum; o Point of Inflection. Example : Using the above example, determine the nature of the Turning Points. Solution: x 0 ~ ~.54 3 f '( x) Slope Nature of T.P Consider the 3 graphs: y x y x x y x x o Graphs similar but different number of stationary points. Ex10C 1 LHS,, 3, 5, 7, 10; Ex10D 1 cef, adf, 4, 10, 1, 13, 17, 18,, 4, 5, 6

25 Maxima/Minima Problems Solving a maximum/minimum problem SETP 1: Need to Maximise/minimise what? Call it A STEP : Write the formula for A =, making up variables where necessary, maybe a diagram could help. STEP 3: Can you write another equation? A = must be written as A = (with only 1 variable on the Right Hand Side). da da STEP 4: Differentiate A = (i.e. ) and equate to zero, 0 and solve for x. STEP 5: Test for the type of stationary point obtained. x da Are any answers impossible (i.e. a negative length) STEP 6: Answer the question in words. Example 1: Four square corners are removed from a sheet of card of dimensions 1cm by 30 cm. The sheet is folded to form an open rectangular container. Find the dimensions (to 1 d.p.), such that the total volume of the container is a maximum. Solution: (using the 6 steps from the photocopy sheet). 1. Need to maximise the of the container,.. H 1 cm H 30 cm L W V = x x 3. Right Hand side has 3 variables ( must be in terms of only 1 variable) L = W = V =

26 4. Maximise let the derivative = 0 i.e. 0. Expand V = 5. (a) (b) H dv dh Therefore a local 6. The maximum/minimum value of a function DOES NOT NECESSARILY OCCUR AT A TURNING POINT. It depends on the feasible Domain caused by the Physical constraints. Ex10F 1,, 3, 4, 6, 7, 1, 14, 16, 17 Worksheet

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28 Absolute Maximum/Minimum Problems Example: Let A be the function that models the total enclosed area when a 100 cm piece of wire is cut into two pieces, where one piece is used to form the perimeter of a square, and the other piece is used to form the circumference of a circle. x (100 x) (a) Show that A can be modelled by, A: 0,100 R, A( x), where x 16 4 cm is the length of the piece of wire used to form the perimeter of the square. (b) For what values of x, is A is a maximum and a minimum? (c) What is the minimum area? Solution: Ex10E 1, 3, 5, 7, 9, 10, 11, 13, 14, 16

29 Families of functions Example 1: Consider the family of functions of the form f ( x) ( x a) ( x b), where a and b are positive constants with b > a. a Find the derivative of f (x) with respect to x. b Find the coordinates of the stationary points of the graph of y f (x). c Show that the stationary point at (a, 0) is always a local minimum. d Find the values of a and b if the stationary points occur where x = 3 and x = 4. 3 Example : The graph of y x 3x, is translated by a units in the positive direction of the x-axis and b units in the positive direction of the y-axis (where a and b are positive constants). 3 a Find the coordinates of the turning points of the graph y x 3x. b Find the coordinates of the stationary points of its image. Ex10G 1,, 3, 5, 7, 9

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