Alan

"Consider all the points in the plane that solve the equation x^2 + 2y^2 = 16. Find the maximum value of the product xy on this graph"

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For a quadratic to have exactly one solution the discriminant must be zero.  In this case that means we must have n^2 - 4*4*25 = 0, which means the positive value of n is 20.

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"In square ABCD, E is the midpoint of line BC, and F is the midpoint of line CD. Let G be the intersection of line AE and line BF. Prove that DG = AB."

Here's a coordinate geometry approach:

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Like so:

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Your expression gives the velocity integral. To get the average, Divide this by t₁-t₀

i.e. v_ave = 4.9(t₀²-t₁²)/(t₁-t₀) → 4.9(t₀-t₁)(t₀+t₁)/(t₁-t₀) → -4.9(t₀+t₁) m/s

The blue graph crosses the line y = 0 twice; once between -1 and -2 And once between 0 and 1.

Perhaps this will help:

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An infinite decimal expansion means there is no end to the 9s.

Let x = 0.999999........ to infinity

Then 10x = 9.999999....... also to infinity

Subtract x from 10x.  The infinite number of 9s to the right of the decimal points cancel and we have:

9x = 9

Hence x = 1.   i.e.   1 = 0.99999............ to infinity

Perhaps these graphs will help:

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The probability that none get it right is q = (1-0.25)*(1-0.5)*(1-0.6)*(1-0.8) or q = (3/4)*(1/2)*(4/10)*(2/10) → 3/100

Hence the probability that at least one gets it right is p = 1 - q or p = 1 - 3/100 → 97/100

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