Part (a)
Use the slope formula to compute the slope from x = 4 to x = 6
So effectively we're finding the slope of the line through (4,70) and (6,68)
We get the following
m = (y2-y1)/(x2-x1)
m = (68-70)/(6-4)
m = -2/2
m = -1
Repeat for the points that correspond to x = 6 and x = 8
m = (y2-y1)/(x2-x1)
m = (73-68)/(8-6)
m = 5/2
m = 2.5
Now average the two slope values
We'll add up the results and divide by 2
(-1+2.5)/2 = 1.5/2 = 0.75
The estimate of T'(6) is 0.75
This works because T'(x) measures the slope of the tangent line on the T(x) curve. Averaging the secant slopes near x = 6 will help give us an estimate of T'(6), which is the slope of the tangent at x = 6 on T(x).
<h3>Answer: 0.75</h3>
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Part (b)
The value T'(6) = 0.75 represents the instantaneous rate of change of the temperature per hour.
More specifically, T'(6) = 0.75 means the temperature is increasing by an estimated 0.75 degrees per hour at the exact instant of x = 6 hours. This instantaneous rate of change is like a snapshot at this very moment in time; in contrast, the slope formula results we computed above measure the average rate of change between the endpoints mentioned.
Answer:
12 percent
Step-by-step explanation:
2.5 ÷ 100 = 0.025
2.8 - 2.5 = 0.3
0.3 ÷ 0.025 = 12
One of the ways to graph this is to use plug in a few x-values and get an idea of the shape. Since the x values keep getting squared, there is an exponential increase on either side of the y-axis. You can see this by plugging in a few values:
When
x=0,f(x)=0
x=1,f(x)=1^2=1
x=2,f(x)=2^2=4
x=3,f(x)=3^2=9
x=4,f(x)=4^2=16
The same holds true for negative x-values to the left of the y-axis since a negative value squared is positive. For example,
x=−1,f(x)=(−1)2=1*−1=1
x=2,f(x)=(−2)2=−2*−2=4
The graph of f(x)=x^2 is called a "Parabola." It looks like this:
