Use a linear approximation (or differentials) to estimate the given number.
e^-0.01
Well, estimating numbers can be a bit like trying to catch a clown on a unicycle; it's not always easy! However, with a little math magic, we can use a linear approximation to estimate the given number e^(-0.01).
Now, when it comes to linear approximations, we start by finding the equation of the tangent line at a given point. In this case, we'll use the point (0, 1) because e^0 is equal to 1.
Now, the derivative of e^x is just e^x itself. So, at the point (0, 1), the slope of the tangent line is e^(0), which is just 1.
Using the point-slope form of a line, y - y₁ = m(x - x₁), we can say that y - 1 = 1(x - 0), which simplifies to y = x + 1.
Now, we can use this linear approximation to estimate e^(-0.01). Plugging in -0.01 for x, we get y = -0.01 + 1, which equals 0.99.
Therefore, using a linear approximation, we estimate that e^(-0.01) is approximately 0.99. But please don't juggle this number too much, it's still an approximation!
To estimate the value of e^-0.01 using linear approximation, we can use the formula:
f(x + Δx) ≈ f(x) + f'(x) * Δx
where f(x) is the function we are interested in, f'(x) is the derivative of the function, Δx is a small change in x, and f(x + Δx) is the estimated value.
In this case, the function we are interested in is f(x) = e^x, and we want to estimate the value of f(x) when x = -0.01.
The derivative of f(x) = e^x is f'(x) = e^x.
Using the formula, we have:
f(-0.01 + Δx) ≈ f(-0.01) + f'(-0.01) * Δx
Since Δx is a small change, let's take Δx = 0.001.
Plugging in the values, we get:
f(-0.01 + 0.001) ≈ f(-0.01) + f'(-0.01) * 0.001
Now, let's calculate the values:
f(-0.009) ≈ f(-0.01) + f'(-0.01) * 0.001
Using the exponential property, we find that f(-0.009) ≈ e^-0.01 * e^0.001.
Calculating this, we get:
f(-0.009) ≈ e^-0.01 * e^0.001 ≈ 0.9900498337
Therefore, the linear approximation for e^-0.01 is approximately 0.9900498337.
To estimate the value of e^-0.01 using linear approximation or differentials method, we can start by considering the linear approximation formula:
L(x) = f(a) + f'(a)(x - a)
where L(x) is the linear approximation of the function, f(a) is the value of the function at point a, f'(a) is the derivative of the function evaluated at point a, and x is the independent variable.
In this case, we have f(x) = e^x and we want to estimate f(-0.01).
Step 1: Find the derivative of f(x)
The derivative of f(x) = e^x is f'(x) = e^x.
Step 2: Choose a value for a
Let's choose a = 0, which is a common choice for linear approximation problems.
Step 3: Evaluate f(a) and f'(a)
f(0) = e^0 = 1
f'(0) = e^0 = 1
Step 4: Plug the values into the linear approximation formula
L(x) = f(a) + f'(a)(x - a)
L(-0.01) = f(0) + f'(0)(-0.01 - 0)
= 1 + 1(-0.01)
= 1 - 0.01
= 0.99
Therefore, using the linear approximation or differentials method, we estimate that e^-0.01 is approximately 0.99.