Suppose that f(t)is a function for which f(80)=8.7and f'(80)=-2.2. Use this information to estimate the following.

Round your answer to one decimal place.

f(80-1)

Well, if f(80) equals 8.7, then f(80-1) would be like the younger sibling of f(80). Let's call it f(79).

Now, since f'(80) is -2.2, we can say that the function f is sloping downwards at that point. So, let's use that information along with our imagination to estimate f(79).

Imagine f(80) is at the top of a roller coaster and f(79) is just one step before that. Since f is sloping downwards, we can assume that f(79) would be slightly lower than f(80). Let's go ahead and estimate it.

Considering the sloping nature of the function, let's say f(79) is roughly 2 units lower than f(80). So, f(79) ≈ 8.7 - 2 = 6.7.

Therefore, our estimate for f(80-1) or f(79) is approximately 6.7.

To estimate f(80-1), we can use the first-order approximation formula:

f(x) ≈ f(a) + f'(a)(x - a)

Given that a = 80, f(a) = 8.7, and f'(a) = -2.2, we can substitute these values into the formula:

f(80-1) ≈ f(80) + f'(80)(80-1 - 80)

Simplifying the equation:

f(80-1) ≈ f(80) + f'(80)(-1)

f(80-1) ≈ 8.7 + (-2.2)(-1)

f(80-1) ≈ 8.7 - 2.2

f(80-1) ≈ 6.5

Therefore, f(80-1) is approximately equal to 6.5.

To estimate the value of f(80-1), we can use the information given about f(t) and its derivative. Here's how we can proceed:

1. We know that f(80) = 8.7, which means that at t = 80, the value of the function is 8.7.

2. We also know that f'(80) = -2.2, which is the value of the derivative of f(t) at t = 80.

3. The derivative of a function gives us the rate at which the function is changing at a particular point. In this case, f'(80) = -2.2 tells us that at t = 80, the function is decreasing at a rate of 2.2 units per unit of t.

4. Since we want to estimate the value of f(80-1), we are looking for the value of the function one unit of t before t = 80.

5. Knowing that the function is decreasing at a rate of 2.2 units per unit of t, we can estimate that the function will decrease by approximately 2.2 units when we move one unit of t before t = 80.

6. Therefore, we can estimate that f(80-1) is approximately 8.7 - 2.2 = 6.5.

So, the estimated value of f(80-1) is 6.5.