Answered

47. If the cost of a certain product in Birr as a function of time in days is given by [tex]$f(t)=3t+t^2$[/tex], then what is the average rate of change over the interval [tex]$2 \leq t \leq 6$[/tex]?
A. 32
B. 10
C. 11
D. 28

48. Let [tex][tex]$f(x)$[/tex][/tex] be a function. Simplify [tex]$\frac{f(x)-f(2)}{x-2}$[/tex] and then evaluate the expression as [tex]$x$[/tex] approaches 2. What is the result?
A. The equation of the secant line to the graph of [tex][tex]$f$[/tex][/tex] over [tex]$[0,2]$[/tex].
B. The gradient of the normal line to the graph of [tex]$f$[/tex] at [tex][tex]$x=2$[/tex][/tex].
C. The equation of the tangent line to the graph of [tex]$f$[/tex] at [tex]$x=2$[/tex].
D. The gradient of the tangent line to the graph of [tex][tex]$f$[/tex][/tex] at [tex]$x=2$[/tex].

49. What is the approximate area of the region enclosed by the graph of [tex]$f(x)=3x^2+1$[/tex], the [tex][tex]$x$[/tex][/tex]-axis, [tex]$x=0$[/tex], and [tex]$x=4$[/tex], where [tex][tex]$[0,4]$[/tex][/tex] is divided into 4 subintervals?
A. 46
B. 94
C. 49
D. 184



Answer :

GinnyAnswer
Let's focus on solving question number 47 in detail.

The given problem is a function that represents the cost of a product in Birr as a function of time in days: [tex]\( f(t) = 3t + t^2 \)[/tex]. The task is to find the average rate of change of this function over the interval [tex]\([2, 6]\)[/tex].

### Step-by-Step Solution:
1. Identify the interval for [tex]\( t \)[/tex]: The interval given is from [tex]\( t = 2 \)[/tex] to [tex]\( t = 6 \)[/tex].

2. Calculate [tex]\( f(t) \)[/tex] for [tex]\( t = 2 \)[/tex]:
[tex]\[ f(2) = 3(2) + 2^2 = 6 + 4 = 10 \][/tex]

3. Calculate [tex]\( f(t) \)[/tex] for [tex]\( t = 6 \)[/tex]:
[tex]\[ f(6) = 3(6) + 6^2 = 18 + 36 = 54 \][/tex]

4. Find the average rate of change: The average rate of change of the function over the interval [tex]\([2, 6]\)[/tex] can be found using the formula:
[tex]\[ \text{Average rate of change} = \frac{f(t_2) - f(t_1)}{t_2 - t_1} \][/tex]
Here, [tex]\( t_1 = 2 \)[/tex] and [tex]\( t_2 = 6 \)[/tex].

5. Substitute the values into the formula:
[tex]\[ \text{Average rate of change} = \frac{f(6) - f(2)}{6 - 2} = \frac{54 - 10}{6 - 2} = \frac{44}{4} = 11 \][/tex]

So, the average rate of change of the cost function over the interval [tex]\([2, 6]\)[/tex] is [tex]\(\boxed{11}\)[/tex].

### Answer:
Therefore, the correct answer to this question is C. 11.