Question

Consider the functions f and g in the tables below. f(x) = 90x2 + 180x + 92 x y 0 92 1 362 2 812 3 1,442 4 2,252 5 3,242 g(x) = 6x x y 0 1 1 6 2 36 3 216 4 1,296 5 7,776 Which of the following statements is true? A. At approximately x = 4.39, the rate of change of f is equal to the rate of change of g. B. As x increases, the rate of change of g exceeds the rate of change of f. C. As x increases, the rate of change of f exceeds the rate of change of g. D. For every value of x, the rate of change of g exceeds the rate of change of f.

Answer 1

As x increases, the rate of change of g exceeds the rate of change of f.

Explanation:

Answer 2

As x increases, the rate of change of g exceeds the rate of change of f.

Explanation:

Given

`f(x) = 90x^2 + 180x + 92`

`\begin{array}{ccccccc}x & {0} & {1} & {2} & {3} & {4} & {5} & f(x) & {92} & {362} & {812} & {1442} & {2252} & {3242} \ \end{array}`

`g(x) = 6^x`

`\begin{array}{ccccccc}x & {0} & {1} & {2} & {3} & {4} & {5} & g(x) & {1} & {6} & {36} & {216} & {1296} & {7776} \ \end{array}`

Required

Which of the options is true?

A. At
`x \approx 4.39`, f(x) has the same rate of change as g(x)

Rate of change is calculated as:

`m = (y_2 - y_1)/(x_2 - x_1)`

For f(x)

`f(x) = 90x^2 + 180x + 92`

`f(4.39) = 90*4.39^2 + 180*4.39 + 92 = 2616.689`

So, the rate of change is:

`m = (2616.689)/(4.39) = 596.06`

For g(x)

`g(x) = 6^x`

`g(4.39) = 6^(4.39) = 2606.66`

So, the rate of change is:

`m = (2606.66)/(4.39) = 593.77`

The rate of change of both functions are not equal at x = 4.39. Hence, (a) is false.

B. Rate of change of g(x) is greater than f(x) with increment in x

Using the formula in (a), we have:

`\begin{array}{ccccccc}x & {0} & {1} & {2} & {3} & {4} & {5} & f(x) & {92} & {362} & {812} & {1442} & {2252} & {3242} & m &\infty & 362 & 406 & 480 & 563 &648.4\ \end{array}`

`\begin{array}{ccccccc}x & {0} & {1} & {2} & {3} & {4} & {5} & g(x) & {1} & {6} & {36} & {216} & {1296} & {7776} & m & \infty & 6 & 18 & 72 & 324 & 1555 \ \end{array}`

From x = 1 to 4, the rate of change of f is greater than the rate of g.

However, from x = 5, the rate of change of g is greater than the rate of f.

This means that (b) is true.

The above table further shows that (c) and (d) are false.