4. Applications of Derivatives

Linear approximation Find the linear approximation to the following functions at the given point a.

f(x) = 4x² + x; a = 1

Linear approximation Find the linear approximation to the following functions at the given point a.

g(t) = √(2t + 9); a = -4

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

e⁰·⁰⁶

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.

1/³√510

45–46. Linear approximation

a. Find the linear approximation to f at the given point a.

b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?

ƒ(x) = x²⸍³ ; a =27; ƒ(29)

Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 - 16t², where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 ≤ t ≤ 5.7 (recall that Δh ≈ h' (a) Δt).

Estimating speed Use the linear approximation given in Example 1 to answer the following questions.

If you travel one mile in 59 seconds, what is your approximate average speed? What is your exact speed?

Find the linearizations of

a. tan x at x = -π/4

Graph the curves and linearizations together.

Find the linearization of ƒ(x) = √(1 + x) + sin x - 0.5 at x = 0.

____

Find the linearization of ƒ(x) = 2/ (1 - x) + √1 + x - 3.1 at x = 0.

Finding Linearizations

In Exercises 1–5, find the linearization L(x) of f(x) at x = a.

f(x) = ∛x, a = −8

Common linear approximations at x = 0 Find the linearizations of the following functions at x = 0.

b. cos x

Linearization for Approximation

In Exercises 7–12, find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.

f(x) = ∛x, a = 8.5

Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.

a. f(x) = (1 − x)⁶

Use the linear approximation (1 + x)ᵏ ≈ 1 + kx to find an approximation for the function f(x) for values of x near zero.

c. f(x) = 1/√(1 + x)