Use a linear approximation (or differentials) to estimate the given number.
(1.999)^4
f(x) = f(a) + (x-a) f'(a)
f(x) = x^4
df/dx = 4x^3
let a = 2
then f(a) = 2^4 = 16
f'(a) = 4*8 = 32
f(x) = 16 + (x-a)(32)
x-a = - .001
so
f(1.999) = 16 -.001(32) = 16 - .032
f(1.999) = 15.968
with calculator it is 15.968 also
Well, my mathematical circus skills might come in handy for this one! Let's estimate the value of (1.999)^4 using a linear approximation, or what I like to call "mathematical tightrope walking."
To begin, we'll choose a number to approximate with that's close to our original number, in this case, 2. Let's use 2 as our approximation.
Now, let's consider the function f(x) = x^4 and find its derivative. The derivative of f(x) with respect to x is 4x^3.
Using our approximation, we can say that f(1.999) is approximately f(2) + f'(2) * (1.999 - 2). In other words:
(1.999)^4 ≈ 2^4 + 4(2^3)(1.999 - 2).
Now, let's calculate the expression:
2^4 = 16,
4(2^3) = 32,
(1.999 - 2) = -0.001.
So, plugging in these values:
(1.999)^4 ≈ 16 + 32 * (-0.001).
Doing the math, we have:
(1.999)^4 ≈ 16 - 0.032.
Therefore, using the linear approximation method, we estimate that (1.999)^4 ≈ 15.968.
Remember, though, this is just a rough estimate, like when a clown tries to juggle apples but ends up dropping some. It's always good to double-check your calculations with more accurate methods!
To estimate the value of (1.999)^4 using linear approximation or differentials, we can start by taking the derivative of the function f(x) = x^4.
f'(x) = 4x^3
Next, we can choose a value close to 1.999 to evaluate the derivative. Let's use x = 2.
f'(2) = 4(2)^3
= 4(8)
= 32
Now, we have the slope of the tangent line to the function f(x) = x^4 at x = 2.
To estimate the value of (1.999)^4, we can use the linear approximation formula:
f(x) = f(a) + f'(a)(x - a)
where a is the value we chose (in this case, a = 2), and x is the value we want to estimate (x = 1.999).
Plugging in the values:
f(1.999) ≈ f(2) + f'(2)(1.999 - 2)
≈ 2^4 + 32(1.999 - 2)
≈ 16 + 32(-0.001)
≈ 16 - 0.032
≈ 15.968
Therefore, using linear approximation, (1.999)^4 is approximately equal to 15.968.
To estimate the value of (1.999)^4 using a linear approximation or differentials, we can start by finding the linear approximation of a function f(x) around a nearby point.
The linear approximation of a function f(x) around a point x=a is given by the equation:
L(x) = f(a) + f'(a)(x - a)
Here, f(a) is the value of the function at the point a, f'(a) is the derivative of the function at the point a, (x - a) represents the change in x from the point a, and L(x) is the linear approximation of the function.
In our case, the function f(x) = x^4 and the point we want to approximate around is a = 2. We will find the linear approximation of f(x) around x = 2.
Step 1: Find f(a) and f'(a)
f(a) = (2)^4 = 16
To find f'(a), we need to find the derivative of f(x) with respect to x and evaluate it at x = a. The derivative of f(x) = x^4 is:
f'(x) = 4x^3
f'(a) = f'(2) = 4(2)^3 = 32
Step 2: Substitute the values into the linear approximation equation
L(x) = f(a) + f'(a)(x - a)
L(x) = 16 + 32(x - 2)
Step 3: Evaluate L(x) at the desired value
To estimate (1.999)^4 using the linear approximation, substitute x = 1.999 into the equation:
L(1.999) = 16 + 32(1.999 - 2)
L(1.999) = 16 + 32(-0.001) = 16 - 0.032 = 15.968
Therefore, using the linear approximation or differentials, we estimate that (1.999)^4 is approximately 15.968.