5. Given the function (cxl, x < 2
f(x) =
CX - 3, x > 2
Determine the value of c so that the function will be continuous everywhere.

I need #5 with an explanation please

The function

is piecewise continuous, since both cx ² and cx - 3 are polynomials. f(x) itself is continuous if both pieces meet at the same defined point. In other words, the limits of f(x) as x → 2 from either side have the same value of f (2) = c•2² = 4c.

We have

so in order for f to be continuous, we need

4c = 2c - 3   →   2c = -3   →   c = -3/2

its 4 because 2 times 2 is 4, if you double the number two you get four

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt