is very difficult to solve for the inverse value of y. If the algebra looks difficult, look for an
obvious solution, such as x = 0 or x = 1 or x = −1.
Now we can use the formula for the derivative of the inverse of f(x):
, where f(a) = c. This formula means that we find the
derivative of the inverse of a function at a value a by taking the reciprocal of the derivative and
plugging in the value of x that makes y equal to a.
21. 997
Recall the differential formula that we use for approximating the value of a function: f(x + ∆x)
≈ f(x) + f′(x)∆x. Here we want to approximate the value of (9.99)^3 , so we’ll use f(x) = x^3 with
x = 10 and ∆x = −0.01. First, we need to find f′(x): f′(x) = 3x^2 . Now, we plug into the formula:
f(x + ∆x) ≈ x^3 + 3x^2 ∆x. If we plug in x = 10 and ∆x = −0.01, we get (10 − 0.01)^3 ≈ (10)^3 +
3(10)^2 (−0.01). If we evaluate this, we get (9.99)^3 ≈ 1,000 + 3(10)^2 (−0.01) = 997.
- −1.732 cm^2
Recall the formula that we use when we want to approximate the error in a measurement: dy =
f′(x) dx. Here we want to approximate the decrease in the area of an equilateral triangle when
we know that it has a side of length 10 cm with a decrease of 0.2 cm, where A(x) = (the
area of an equilateral triangle of side x) with dx = −0.2. We find the derivative of the area: A′
(x) = . Now we can plug into the formula: dA = dx. If we plug in x = 10
and dx = −0.2, we get dA = (-0.2) = − ≈ −1.732.
