4. Applications of Derivatives

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

g(t) = √(2t + 9); a = -4

Suppose f is differentiable on (-∞,∞), f(5.99) = 7, and f(6) = 7.002. Use linear approximation to estimate the value of f'(6).

21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].

b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.

ƒ(x) = { - 2x if x < 0 ; x if x ≥ 0 ; [-1, 1]

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?

21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].

b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.

ƒ(x) = 7 -x² ; [-1; 2]

154. The linearization of log₃x

a. Find the linearization of

f(x) = log₃xatx = 3.

Then round its coefficients to two decimal places.

Use linear approximation to estimate f (5.1) given that f(5) = 10 and f'(5) = -2.

Use linear approximation to estimate f (3.85) given that f(4) = 3 and f'(4) = 2.

Suppose f is differentiable on (-∞,∞) and the equation of the line tangent to the graph of f at x = 2 is y = 5x -3. Use the linear approximation to f at x = 2 to approximate f(2.01).

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

e⁰·⁰⁶