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There is an exponential function that is $y$ positive and decreasing. Two coordinates are $(0, 3600)$ and $(2,225)$

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Find derivatives of exponential functions

Full solution

Q. There is an exponential function that is

y

positive and decreasing. Two coordinates are

(0, 3600)

and

(2,225)

Identify General Form: Identify the general form of an exponential function. The general form is $y = ab^x$ , where $a$ and $b$ are constants to be determined.
Find $a$ : Use the first point $(0, 3600)$ to find $a$ . $\newline$ Substitute $x = 0$ and $y = 3600$ into the equation: $\newline$ $3600 = ab^0$ , $\newline$ Since $b^0 = 1$ , we get $3600 = a$ .
Substitute 'a': Substitute 'a' back into the equation. $\newline$ Now the equation is $y = 3600b^x$ .
Find 'b': Use the second point $(2, 225)$ to find 'b'. $\newline$ Substitute $x = 2$ and $y = 225$ into the equation: $\newline$ $225 = 3600b^2$ , $\newline$ Solve for $b$ : $b^2 = \frac{225}{3600}$ , $\newline$ $b^2 = 0.0625$ , $\newline$ $b = \sqrt{0.0625}$ , $\newline$ $b = 0.25$ .

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