Question
Compare the growth rate of the functions f(x) = 4x and g(x) = the square root of the quantity of 16 raised to the x power plus 2 raised to the x power.
a. f(x) grows faster than g(x).
b. f(x) and g(x) grow at the same rate.
c. g(x) grows faster than f(x).
d. It cannot be determined.
a. f(x) grows faster than g(x).
b. f(x) and g(x) grow at the same rate.
c. g(x) grows faster than f(x).
d. It cannot be determined.
Answers
exponentials eventually grow faster than any polynomials
16^x + 2^x grows faster than 16x^2
g = 16^x + 2^x = 2^(4x) + 2^x
g' = ln2 * 5*2^x
let F(x) = (4x)^2
F' = 32x
so, as soon as ln32 * 2^x > 32x g outstrips F
x > 8.32
16^x + 2^x grows faster than 16x^2
g = 16^x + 2^x = 2^(4x) + 2^x
g' = ln2 * 5*2^x
let F(x) = (4x)^2
F' = 32x
so, as soon as ln32 * 2^x > 32x g outstrips F
x > 8.32
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