f(x)=2x^4+5x^3-x^2-3x+1
a)Determine the average slope between the points where x=1 and x=3.
b)Determine the instantaneous slope at each of these points.
slope: 22
instantaneous: 22 1/2
f(1) = 2 + 5 - 1 - 3 + 1 = 4
f(3) = 162 + 135 - 9 - 9 + 1 = 280
for between the points (1,4) and (3,280)
slope = (280-4)/(3-1) = 138
f'(x) = 8x^3 + 15x^2 - 2x - 3
f ' (1) = 8+15 - 2 - 3 = 18
at (1,4) the real slope is 18
f ' (3) = 216 + 135 - 6 - 3 = 342
at (3,280) the slope is 342
To determine the average slope between two points, we need to find the slope of the function between those points. Let's start by finding the slope between x = 1 and x = 3.
a) Average Slope between x=1 and x=3:
To find the average slope between two points, we need to calculate the slope of the function at each point and then find the average of those slopes.
Step 1: Calculate the slope at x = 1:
To find the slope at a specific point, we need to find the derivative of the function and evaluate it at that point. So let's start by finding the derivative of the given function, f(x).
f(x) = 2x^4 + 5x^3 - x^2 - 3x + 1
To find the derivative, we differentiate each term with respect to x:
f'(x) = (8x^3) + (15x^2) - (2x) - 3
Now we need to substitute x = 1 into f'(x) to find the slope at x = 1:
f'(1) = (8(1)^3) + (15(1)^2) - (2(1)) - 3
= 8 + 15 - 2 - 3
= 18
So, the slope at x = 1 is 18.
Step 2: Calculate the slope at x = 3:
Similarly, we substitute x = 3 into f'(x) to find the slope at x = 3:
f'(3) = (8(3)^3) + (15(3)^2) - (2(3)) - 3
= 8(27) + 15(9) - 6 - 3
= 216 + 135 - 6 - 3
= 342
So, the slope at x = 3 is 342.
Step 3: Calculate the average slope:
To find the average slope between x = 1 and x = 3, we need to calculate the average of these slopes.
Average Slope = (slope at x = 1 + slope at x = 3) / 2
= (18 + 342) / 2
= 360 / 2
= 180
Therefore, the average slope between x = 1 and x = 3 is 180.
b) Instantaneous Slope at each point:
To find the instantaneous slope at each point, we need to find the derivative of the function at that specific point.
At x = 1:
We have already found the derivative of the function, f'(x), in Step 1 of part a). So, f'(1) = 18. Therefore, the instantaneous slope at x = 1 is 18.
At x = 3:
Similarly, f'(3) = 342. So, the instantaneous slope at x = 3 is 342.
Therefore, the instantaneous slope at x = 1 is 18, and the instantaneous slope at x = 3 is 342.