Gradient of a curve and equation of a

Gradient of a curve and equation of a

• 1.
  Gradient of aCurve and Equation of a Tangent to a Curve Math Studies 2 – Topic 7
• 2.
  Find the gradientof the tangent to the curve whose 4 equation is y = + 2 at the point where x=2. x −1 y = 4x + 2 substitute f ' ( x ) = −4 x + 0 −2 −4 f ( x) = 2 = −1 2 4 f ' ( x ) = −4 x −2 = 2 x
• 3.
  A power boatmoves in a straight line such that at time t seconds its distance s from a fixed point O on that line is given by s = 2t − 3t + 1 2 Find the speed after 3 seconds. ds substitute = 4t − 3 t =3 dt speed = 4(3) − 3 = 9m / s
• 4.
  Find the valueof the gradient of the curve whose equation is y=(x-3)(x+2) at the point where it crosses the positive x-axis (x=3). multiply substitute y = x − x−6 2 x=3 f ' ( x) = 2(3) − 1 = 5 f ' ( x) = 2 x − 1
• 5.
  Think Back Whatare the steps I used to solve the problems?
• 6.
  Steps to findingthe gradient of any curve • Differentiate to find the gradient • Substitute the particular value of x to find the gradient of the curve at that point
• 7.
  Find the equationof the tangent to the curve whose 2 equation is y = 3x 2 + − 5 at the point P where x=2. x dy 2 2 −2 = 6x − 2x = 6x − 2 y = 3(2) 2 + −5 = 8 dx x 2 y =8 P(2,8) substitute dy 2 1 y = mx + c = 12 − = 11 dx 4 2 1 1 8 = 11 (2) + c m = 11 2 2 c = 15
• 8.
  equation 1 y = 11 x − 15 2
• 9.
  Think Back Whatare the steps I used to solve the problems?
• 10.
  Equation of theTangent at a Given Point • Find the gradient at the point P and call it m • Find the y-coordinate of P as well as the x- coordinate (sometimes it will be given) • Use y=mx+c and substitute for m, x, and y at the point P • Write the equation with values of m and c
• 11.
  Independent Practice
• 12.
  1) Find theequation of the tangent to the curve whose 4 equation is y=x²-2x +3 at the point where x=1. Show all working. 2) Find the equation of a tangent to the curve whose equation is y=2x³-4x at the point where x=2. 3) The curve whose equation is y=x²-4x+15 has gradient 6 when x=a. Find the value of a. 4) The tangent to the parabola given by y=x²+3x-8 has gradient 7 at the point P. Find the coordinates of P.
• 13.
  Solution 1)y=2x+4 2)y=20x-32 3)5 4)(2,2)
• 14.
  Resources www.info.oxford-consulting.com www.wclipart.com www.clipartspot.net Mathematical Studies CourseCompanion (Oxford)