Tutor profile: Jawwad S.
Find the turning point of the following curve and determine whether it is a maximum or minimum: f(X)=3Xsqr +12X+9=0
We know that in the turning point, the gradient of the curve is zero Taking the first derivatives : (d/dx)(3Xsqr+12X+9) 6X+12=0 6X=-12 X=-2 Since the gradient of the curve (first derivative) of the curve is zero, this must be the turning point. The y-coordinate must be f(-2)=-3 therefore, turning point=(-2,-3) Now, Taking the second derivative: (d/dx)(6X+12)=6 (which is a +ve number) Therefore, the turning point must be a minimum. ANSWER: The curve has a minimum point at (-2,-3).
A long cylinder is made by joining two cylinders, of identical dimensions, end-to-end, but one of which was made out of copper and the other was made out of wood. The new cylinder is wrapped in a single layer of white paper and held over a flame for about 5 seconds. It is observed that only half of the paper covering the long cylinder is blackened, while the other half remains relatively unsoiled. Which side do you think was blackened? explain your reasoning. A simplified representation of the observation is given below: (3 points) (charred side) [|||||||||||||||||||||||||||__________] ( unsoiled side)
I think that the paper covering the wooden cylinder was blackened. The observation is most likely caused by the difference in the thermal conductivity of the two materials. When held over a flame, the heat was quickly carried away from the part of the paper covering the copper cylinder, by the process of thermal conduction; unlike the part that covered the wooden cylinder. Thus, heat could build up on one side quicker than the other, initiating combustion, and blackening the paper. Mark Scheme: First point- For identifying that the wooden side will blacken. Second point- For identifying the underlying physics of thermal conductivity as responsible. Third point- For explaining how the difference in thermal conductivity leads to the observed results.
Solve the following pair of simultaneous equations: 2x+y=7-----equation (1) 3x-y=9-----equation (2)
From equation (1): 2x+y=7 y=7-2x -----equation (3) Substituting equation (3) into equation (2): 3x-(7-2x)=9 3x-7+2x=9 5x-7=9 5x=9+7 x=16/5 Substituting the value of x into equation (3): y=7-2*(16/5) y=3/5 ANSWER: x=16/5 and y=3/5.
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