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Tam 2nd

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Tam 2nd

  1. 1. Derivative Application <ul><li>Group Members: </li></ul><ul><li>Baba Qazi; Joey Lee; Tam Dong </li></ul>
  2. 2. Tangent line equation <ul><li>y=m(x-x 1 )+y 1 </li></ul>What do you need to find a tangent line equation? m= Slope of the tangent line (x,y): coordinate of a point (x 1 ,y 1 ): coordinate of another point The slope and a point
  3. 3. <ul><li>Ex: If f(x)=x 2 +2. Determine the slope of the tangent line to f(x) at x=5. </li></ul><ul><li>1. f(x)=x 2 +2 =>f’(x)=2x => f’(5)=10 </li></ul><ul><li>2. f(5)=25+2=27 </li></ul><ul><li>3. y=m(x-x 1 )+y 1 </li></ul><ul><li>4. y=10(x-5)+10 </li></ul>Derivative Applications of tangent line
  4. 4. Derivative Applications of tangent line (cont) <ul><li>Ex: If f(x)=sin(x)cos(x). Determine the slope of the tangent line to f(x) at x=π/4. </li></ul><ul><li>1.f(x)=sin(x)cos(x) => product rule </li></ul><ul><li>2.f(x)=sin(x);f’(x)=cos(x) | g(x)=cos(x);g’(x)=-sin(x) </li></ul><ul><li>3.f’(x)= -sin(x)sin(x)+2cos(x) = 2cos(x)-2sin(x) </li></ul><ul><li>4.f(π/4)=sin(π/4)cos(π/4)=(√2/2)(√2/2)=1 </li></ul><ul><li>5.f’(π/4)=2cos(π/4)-2sin(π/4)=2(√2/2) - 2(√2/2)=0 </li></ul><ul><li>6.y=m(x-x 1 )+y 1 => y=0[x-(π/4)]+1 </li></ul>
  5. 5. Slope at a point <ul><li>The derivative of the function at a point is the slope of the line tangent to the curve at the point, and is thus equal to the rate of change of the function at that point. </li></ul>
  6. 6. Slope at a point <ul><li>Ex: Find the instantaneous rate of change of y=6x 2 when x=3 </li></ul><ul><li>d/dx=12x => f’(3) = 12.3= 36 </li></ul>
  7. 7. Approximation <ul><li>Approximation is used to determine the slope of the equation by (∆y/∆x) </li></ul><ul><li>Examples </li></ul><ul><li>Forward quotient: (8-6)(3-2)= 2 </li></ul><ul><li>Backward quotient: (6-4)(2-1)= 2 </li></ul><ul><li>Symmetric quotient: (8-4)(3-1)= 2 </li></ul>x 1 2 3 y 4 6 8
  8. 8. Implicit Differentiation <ul><li>The purpose of using implicit differentiation is when it is very difficult to express y as a function respect to x </li></ul>Derive: y 3 + y 2 - 5y -x 2 = -4 1. y 3 + y 2 - 5y -x 2 = -4 2. (d/dx) [y 3 + y 2 - 5y -x 2 ] = (d/dx) -4 3. 3y 2 (dy/dx) + 2y(dy/dx) - 5(dy/dx) - 2x = 0 4. (dy/dx)(3y 2 + 2y -5) = 2x 5. (dy/dx) = 2x/(3y 2 + 2y -5)
  9. 9. Guidelines for Implicit Differentiation <ul><li>1. Differentiate both sides of the equation with respect to x. </li></ul><ul><li>2. Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation. </li></ul><ul><li>3. Factor dy/dx out of the left side of the equation </li></ul><ul><li>4. Solve for dy/dx </li></ul>

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