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### The points P (2,3.6) and Q(2.2,2.4) lie on the curve y=f(x) . Use P and Q to estimate the gradient of the curve at the point where x=2 .

To answer this question you will need to recall the formula for the gradient of a straight line. Cordinates are written in the form (x​1,x2​) and (y1,​y2​). The formula takes into consideratoon the cordinates of two points on a curve or line joined by a straight line. The formula is as follows : (y2 -y​1 )/(x2-x1​). By substituting in the cordinates in appropriatley we end up with and equation as follows, (2.4-3.6)/(2.2-2). This will give an answer of  -6. Another way of aproaching this question would be to look at the equation of a straight line, y=mx+c,where c is the y intercept and m the gradient. We can see that is we substitute in the number from both points into this formla we end up with two equations with two unknowns. 1)3.6=2m+c and 2) 2.4=2.2m+c. We know that two equations with two unkowns means we can solve this problem using simultaneous equations. therfore re can rearange both equations to equal c. We end up with 1) 3.6-2m=c and 2) 2.4-2.2m=c . We put them togther; 3.6-2m=2.4-2.2m. We bring like terms to the same side 3.6-2.4=-.2.2m+2m. We then solve both sides to give 1.2=-0.2m. To find m we rearage to find my and end up with 1.2/-0.2=m. This gives us m=-6. This second method gives the same answer as the first and will be a good was of solving equations of this type when one cordinate is missing but the y intercept is known.

1 year ago

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