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Symbol Defintion
Example : x^2 means x square
CA 02 01.Definition of derivative
y' is the first derivative of y
- Definition : y' = Lim[(F(x+h) - F(x))/h] when h goes to zero
- Let y = F(x)
- Let point A be (x, y1) and point B be (x+h, y2)
- (F(x+h) - f(x))/h is the slope of line AB
- If B is very close to A, then (F(x+h) - f(x))/h is slope of line at point A
- The line is the tangent of the graph of y = F(x) at point B
- Hence we define y' = Lim[(F(x+h) - F(x))/h]
- y' is the first derivative of y
- y' is also called the differentiation of y
- y' is the slope of the curve of y = F(x)
- y' = dy/dx
- y' = Dx(y)
- y' = F'(x) if y = F(x)
- It is the slope of tangent to curve y = F(x)
- If tangent to curve making angle A with x-axis
- Then y' = Slope = tan(A)
- It is the change rate in motion with equation d = F(t)
- The speed at given time t is V = d/dt(F(t))
- It is different from then mean speed between two given time
- It is y" = dy'/dx
- It can also use F"(x)
- Application of 2nd derivative
- For graph, it define the concavity
- For motion, it is the acceleration
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CA 02 02. Power rule
If y = x then y' = 1
- y' = Lim[((x+h)-x)/h] = Lim[h/h] = 1 as h goes to zero
- y' = Lim[((x+h)^2 - x^2)/h]
- y' = Lim[(x^2 + 2*x*h + h^2 - x^2)/h]
- y' = Lim[(2*h*x + h^2)/h]
- y' = Lim[(2*x + h)]
- y' = 2*x if h goes to zero
- y' = Lim[((x+h)^3 - x^3)/h]
- y' = Lim[(x^3 + 3*(x^2)*h + 3*x*h^2 + h^3 - x^3)/h]
- y' = Lim[(3*h*x^2 + 3*x*h^2 + h^3)/h]
- y' = Lim[(3*x + 3*x*h + h^2)]
- y' = 3*x^2 if h goes to zero
- Since y = x^3 and y' = 3*x^2
- Since y = x^2 and y' = 2*x
- Since y = x^1 and y' = 1
- Hence y = x^n, then y' = n*x^(n-1)
- If y = c, the y' = 0 where c is constant
- This rule can be applied to negative n
- y = x*(-1), y' = -1*x^(-2)
- y = x*(-2), y' = -2*x^(-3)
- y = x*(-3), y' = -3*x^(-4)
- This rule can be applied to negative 1/n
- y = x*(-1/2), y' = (-1/2)*x^(-3/2)
- y = x*(-1/3), y' = (-1/3)*x^(-4/3)
- y = x*(-1/4), y' = (-1/4)*x^(-5/4)
- y' = 2*x - 6
- y" = 2
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CA 02 03. Derivative and curve
It change the shape of the curve. Example
- y = x^2 is parabola
- y' = 2*x is a line
- 1. If y' is greater than zero, the curve is increasing
- 2. If y' is less than zero, the curve is decreasing
- 3. If y' = 0, the curve has maximum point
- If y' = 0 at x = a
- If y' LT 0 when x LT a. Curve decreases
- If y' GT 0 when x GT a. Curve increases
- Hence curve is concave upward
- 4. If y' = 0, the curve has minimum point
- If y' = 0 at x = a
- If y' LT 0 when x GT a. Curve increases
- If y' GT 0 when x LT a. Curve decreases
- Hence curve is concave downward
- 4. If y' = 0, the curve has a point of inflexion
- If y' = 0 at x = a
- If y' GT 0 when x GT a. Curve increases
- If y' GT 0 when x LT a. Curve increases
- Hence the curve is concave upward when x lT a and downward when x GT a
- Or the curve is concave downward when x lT a and upward when x GT a
- It is in 1st quadrant (X GT 0 and y GT 0)
- The curve is increasing (X GT 0 and y' GT 0)
- the curve is concave downward or concave upward
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Compare graphs of y = F(x) and y = F'(x)
Program 02 01
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CA 02 04. Second derivative and curve
The minimum point
- y' = 0 when x = a
- y" = + when x = a
- y' = 0 when x = a
- y" = - when x = a
- y' = 0 when x = a
- y" = 0 when x = a
- It is in 1st quadrant (X GT 0 and y GT 0)
- The curve is increasing (y' GT 0)
- the curve is concave upward (y" GT 0)
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CA 02 05. Produc rule
Rule
- If y = F(x)*G(x), then y' = F'(x)*G(x) + F(x)*G'(x)
- If we use power rule, we have to expand it as polynomial form
- If we use product rule, the question becomes simple
- Let F(x) = (x+1)^3 and G(x) = (x+2)^5
- Hence y' = F'(x)*G(x) + F(x)*G'(x)
- Hence y' = (3*(x+1)^2)*((x+2)^5) + ((x+1)^3)*(5*(x+2)^4)
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CA 02 06. Quotient rule
Rule
- If y = F(x)/G(x), then y' = (F'(x)*G(x) - F(x)*G'(x))/(G(x)^2)
- If we can not use power rule
- If we have to use quotient rule
- Let F(x) = (x-1) and G(x) = (x+2)
- Hence y' = ((F'(x)*G(x) + F(x)*G'(x))/((x+2)^2)
- Hence y' = ((1)*(x+2) + (x-1)*(1))/((x+2)^2)
- If y = x^(-n), then y = 1/(x^n)
- Let F(x) = 1 and G(x) = x^n
- Hence y' = (F'(x))*(G(x)) - (F(x))*(g'(x))/(G(x)^n)
- Hence y' = ((0)*(x^n) - (1)*(n*x^(n-1)))/(x^n)^2)
- Hence y' = -n*(x^((n-1)-2*n))
- Hence y' = -n*(x^(-n-1))
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CA 02 07. Chanin rule
Definition : y = F(u) and find dy/dx
- dy/dx is the derivative of y with respective of function of x
- dy/du is the derivative of y with respective of function of u
- Hence we can put y' = dy/dx as y' = (dy/du)*(du/dx)
- Let u = 1 - x^2 and y = Sqr(u) = u^(1/2)
- Hence dy/dx = (dy/du)*(du/dx)
- Hence dy/dx = (1/2)*(x^(-1/2))*(-2*x)
- Hence dy/dx = (-x)*(x^(-1/2))
- Hence dy/dx = (-x)/Sqr(x)
- Method 1 : Use definition of derivative
- y' = Lim[Sqr(x+h) - Sqr(x))/h]
- Both numeritor and denominator time (Sqr(x+h) + Sqr(x))
- y' = lim[(Sqr(x+h) - Sqr(x))*(Sqr(x+h) + Sqr(x))/(Sqr(x+h) + Sqr(x))*h)
- Use (a-b)*(a+b) = a^2 - b^2
- y' = lim[((x+h) - x)/(Sqr(x+h) + Sqr(x))*h)
- y' = lim[(h)/(Sqr(x+h) + Sqr(x))*h)
- y' = lim[(1)/(Sqr(x+h) + Sqr(x)))
- y' = lim[(1)/(2*Sqr(x))
- y' = 1/(2*Sqr(x))
- Method 2 : Use power rule
- y' = (1/2)*(x^(1/2-1))
- y' = 1/(2*x^(1/2))
- y' = 1/(2*Sqr(x))
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CA 02 08. Power rule to rational power
y = x^(1/2)
- Let n = 1/2
- y' = n*x^(n-1) = (1/2)*(x^(1/2-1)) = (x^(-1/2))/2
- Let n = -2
- y' = n*x^(-2-1) = (-2)*(x^(-2-1)) = -2*(y^(-3))
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CA 02 09. Equation of tangent
Function y = F(x) = x^2 - 6*x + 8. Find tangent to curve when x = 4
- Point on curve is (4, 0)
- Slope s = y' = 2*x - 6
- Slope s = 2*4 - 6 = 2
- Equation of tangent y = s*x + b
- When x = 4 and y = 0, hence 0 = 2*4 + b or b = -8
- Hence equation of tangent is y = 2*x - 8
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CA 02 10. Sketch using y' and y"
Sketch using section domain and sign of range
- We can only guess the concavity of the curve
- We can only estimate the extreme points and point reflexion
- We can find the curve is increasing for give domain section if y' GT 0
- We can find the curve is decreasing for give domain section if y' LT 0
- We can find the extreme point when y' = 0
- We can find the curve is concave upward for give domain section if y" GT 0
- We can find the curve is concave downward for give domain section if y" LT 0
- We can find the point of inflecion poin when y" = 0
- Find y' and y"
- y' = (-4*x)/(1 + x^2)
- y" = ((-4)*(1+x^2) - (-4*x)*(2*x))/((1 + x^2)^2) = (4*x^2 - 4)/((1+x^2)^2)
- Domain sections : -1, 0, 1 (-1, and 1 obtained from y")
- x LT -1 : y GT 0, y' GT 0, y" GT 0
- x EQ -1 : y = 1
- Hence the curve is increasing and concave upward from y = 0 to y = 1
- x GT -1 and x LT 0, y' GT 0, y" LT 0
- x EQ 0 : y = 2
- Hence the curve is increasing and concave downward from y = 1 to y = 2
- x GT 0 and x LT 1, y' LT 0, y" LT 0
- x EQ 1 : y = 1
- Hence the curve is decreasing and concave downward from y = 2 to y = 1
- x GT 1 : y GT 0, y' LT 0, y" GT 0
- x EQ -1 : y = 1
- Hence the curve is decreasing and concave upward from y = 1 to y = 0
- Note : Use values of y we can not get the special properties at x = -1 and x = 1
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Sketch using y' and y"
Program 03 01 : y = 1/(1 + x^2)
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CA 02 11. Sketch curve of function y = F(x)
Proposed information for sketch
- 1. Find domain for existing range
- 2. Find intercepts
- y-intercept = F(0)
- x-intercept : roots of F(x) = 0
- 3. Symmetry
- Odd function : F(-x) = -F(x)
- Even function : F(-x) = +F(x)
- Periodic function : F(x + p) = f(x) and p is the period
- 4. Asymptotes
- Vertical asymptote : x = a if y goes to infinite
- Horizotal asymptote : y = b when x goes to infinite
- Asymptote is function : y = G(x) when x goes to infinite
- 5. Interval of increase of decrease
- Interval of increase : y' GT 0
- Interval of decrease : y' LT 0
- 6. Local maximum or minimum
- Maximum point : y' = 0 and y" = (-)
- Minimum point : y' = 0 and y" = (+)
- 7. Concavity and point of inflection
- Concave upward : y" GT 0
- Concave downward : y" LT 0
- Point of inflection : y" = 0
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CA 02 12. Sketch y = (2*x^2)/(x^2 - 1) using CA 02 11 information
Sketch information
- 1. Domain : all x except x = -1 and x = 1
- 2. Intercepts
- y-intercept : y = 0 when x = 0
- x-intercept : x = 0 when y = 0
- 3. Symmetry
- Since F(-x) = F(x), it is symmetrical to y-axis
- 4. Asymptotes
- Vertical asymptote x = -1
- Vertical asymptote x = +1
- Horizontal asymptote : y = 2
- 5. y' = ((4*x)*(x^2 - 1) - (2*x^2)*(2*x))/(x^2 - 1)^2 = -(4*x)/(x^2 - 1)^2
- y' = -(4*x)/(x^2 - 1)^2
- When x LT -1, y' = (+). Curve increases from y = 2 to y = +infinite at x = -1
- When x between -1 and 0, y' = (+). Curve increases from -infinite to 0
- When x between 0 and 1, y' = (-). Curve decreases from 0 to -infinite
- When x GT 1, y' = (-). Curve decreases from y = +infinite to y = 2
- 6. Maximum or minimum
- When x = 0 and y' = 0. It is maximum
- 7. y" = (-4*(x^2 - 1)^2 + (4*x)*2*(x^2 - 1)*2*x)/(x^2 - 1)^4
- y" = (-4*(x^2 - 1) + 16*x^2)/(x^2 - 1)^3
- y" = 4*(3*x^2 + 1)/(x^2 - 1)^3
- When x LT -1, y" = (+). Curve concave upward
- When x between -1 and 1, y" = (-). Curve concave downward
- When x GT 1, y" = (+). Curve concave upward
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CA 02 00. Outlines
Defintion : y = F(x)
- drievative = dy/dx = Lim[(F(x+h)-F(x))/h] as h goes to zero
- 1st derivative = y' = F'(x) = dy/dx
- 2nd derivative = y" = F"(x) = dy'/dx
- Power rule : y = x^n and dy/dx = n*x^(n-1)
- Product rule : y = F(x)*G(x) and dy/dx = F'(x)*G(x) + F(x)*G'(x)
- Quotient rule : y = F(x)/G(x) and dy/dx = (F'(x)*G(x) - F(x)*G'(x))/(G(x)^2)
- Chain rule : y = F(u) and dy/dx = (dy/du)*(du/dx)
- (a + b)^2 = a^2 + 2*a*b + b^2
- (a + b)^3 = a^3 + 3*(a^2)*b + 3*a*(b^2) + b^3
- sin(A + B) = sin(A)*cos(A) + cos(A)*sin(A)
- Lim[sin(x)/x] = 1 as x goes to 0
- Lim[(exp(x)- 1)/x] = 1 as x goes to 0
- Lim[(1 + x)^(1/x)] = 1 as x goes to 0
- sin(A+B) : in TR 07 00
- Limit : in CA 01 00
- Curve of y = a*x^2 + b*x + c a parabola
- Curve of y' = 2*a*x + b becomes a stright line
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Diagrams
Program 01 02 is y = a*x^2 + b*x + 1
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