Jac and Jill have built a ramp for their toy car. They will release the car at the top of the ramp and the car will jump off the end of the ramp.
The cross-section of the ramp is modelled by the function
The graph of
- Find
for . (2 marks)
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- i. Find the coordinates of the point of inflection of
. (1 mark)
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- ii. Find the interval of
for which the gradient function of the ramp is strictly increasing. (1 mark)
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- iii. Find the interval of
for which the gradient function of the ramp is strictly decreasing. (1 mark)
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Jac and Jill decide to use two trapezoidal supports, each of width
- Determine the value of the ratio of the area of the trapezoidal cross-sections to the exact area contained between
and the -axis between and . Give your answer as a percentage, correct to one decimal place. (3 marks)
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- Referring to the gradient of the curve, explain why a trapezium rule approximation would be greater than the actual cross-sectional area for any interval
, where . (1 mark)
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- Jac and Jill roll the toy car down the ramp and the car jumps off the end of the ramp. The path of the car is modelled by the function
, where
-
is continuous and differentiable at , and is where the car lands on the ground after the jump, such that .
-
- Find the values of
and . (2 marks)
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- Determine the horizontal distance from the end of the ramp to where the car lands. Give your answer in centimetres, correct to two decimal places. (1 mark)
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- Find the values of
a.
b.i
b.ii
b.iii
c.
d.
e.i
e.ii
