A function f(t) increases by a factor of 10 over every unit interval in t and f(0) = 1. Which could be a function rule for f(t)? Choices: (A)f(t) = 1 + t/10 (B)f(t) = 1 + 10t (C)f(t) = 10^t (D)f(t) = 1 - 10^t

Check Option (A): Check option (A) f(t) = 1 + t/10. If t increases by 1, f(t) should increase by a factor of 10. So, f(1) should be 10 * f(0) = 10 * 1 = 10. Calculate f(1) for option (A): f(1) = 1 + 1/10 = 1.1, which is not 10 times f(0).Check Option (B): Check option (B) f(t) = 1 + 10t. Calculate f(1) for option (B): f(1) = 1 + 10 * 1 = 11, which is not 10 times f(0).Check Option (C): Check option (C) f(t) = 10^t. Calculate f(1) for option (C): f(1) = 10^1 = 10, which is 10 times f(0). This looks promising, but let's check if it holds for another value. Calculate f(2) for option (C): f(2) = 10^2 = 100, which is 10 times f(1). This option seems to fit the condition.Check Option (D): Check option (D) f(t) = 1 - 10^t. Calculate f(1) for option (D): f(1) = 1 - 10^1 = -9, which is not 10 times f(0) and does not even increase as t increases.

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A function $f(t)$ increases by a factor of $10$ over every unit interval in $t$ and $f(0) = 1$ . Write a function rule for $f(t)$ .

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Calculus

Intermediate Value Theorem

Full solution

Q. A function

f(t)

increases by a factor of

10

over every unit interval in

t

and

f(0) = 1

. Write a function rule for

f(t)

Check Option (A): Check option (A) $f(t) = 1 + \frac{t}{10}$ . If $t$ increases by $1$ , $f(t)$ should increase by a factor of $10$ . So, $f(1)$ should be $10 \times f(0) = 10 \times 1 = 10$ . Calculate $f(1)$ for option (A): $f(1) = 1 + \frac{1}{10} = 1.1$ , which is not $10$ times $t$ $0$ .
Check Option (B): Check option (B) $f(t) = 1 + 10t$ . Calculate $f(1)$ for option (B): $f(1) = 1 + 10 \times 1 = 11$ , which is not $10$ times $f(0)$ .
Check Option (C): Check option (C) $f(t) = 10^t$ . Calculate $f(1)$ for option (C): $f(1) = 10^1 = 10$ , which is $10$ times $f(0)$ . This looks promising, but let's check if it holds for another value. Calculate $f(2)$ for option (C): $f(2) = 10^2 = 100$ , which is $10$ times $f(1)$ . This option seems to fit the condition.
Check Option (D): Check option (D) $f(t) = 1 - 10^t$ . Calculate $f(1)$ for option (D): $f(1) = 1 - 10^1 = -9$ , which is not $10$ times $f(0)$ and does not even increase as $t$ increases.