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Gradient of a curve and equation of a

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Gradient of a curve and equation of a

1. 1. Gradient of a Curve andEquation of a Tangent to a Curve Math Studies 2 – Topic 7
2. 2. Find the gradient of the tangent to the curve whose 4equation 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. 3. A power boat moves in a straight line such that at time tseconds its distance s from a fixed point O on that line isgiven 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. 4. Find the value of the gradient of the curve whoseequation is y=(x-3)(x+2) at the point where it crosses thepositive 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. 5. Think Back What are thesteps I used to solve the problems?
6. 6. Steps to finding the 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. 7. Find the equation of the tangent to the curve whose 2equation 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. 8. equation 1y = 11 x − 15 2
9. 9. Think Back What are thesteps I used to solve the problems?
10. 10. Equation of the Tangent 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. 11. Independent Practice
12. 12. 1) Find the equation 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. 13. Solution1)y=2x+42)y=20x-323)54)(2,2)
14. 14. Resourceswww.info.oxford-consulting.comwww.wclipart.comwww.clipartspot.netMathematical Studies Course Companion (Oxford)