4. Applications of Derivatives

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = x/6 - sec x on [0,8]

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = x⁵/5 - x³/4 - 1/20

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.

f(x) = x²(x - 100) + 1

Let ƒ(x) = 3x - x³ . Show that the equation ƒ(𝓍) = -4 has a solution in the interval [2,3] and use Newton’s method to find it.

The Mean Value Theorem                                                                                                                                                                  

 a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .

[Technology Exercises] b.Find the solution to as many decimal places as you can.  

As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal/min. (An acre-foot is 43,560 ft³, the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.) 

Calculate the first derivatives of ƒ(𝓍) = 𝓍²/ (𝓍² + 1) and g(𝓍) = ―1/ (𝓍² + 1) . What can you conclude about the graphs of these functions?

[Technology Exercise] Roots

Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.

b. Solve the equation ƒ(𝓍) = 0 graphically with an error of magnitude at most 10⁻⁸ .

[Technology Exercise] Roots

Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.

c. It can be shown that the exact value of the solution in part (b) is

(1/2 + √69/18)¹/³ + (1/2 ― √69/18)¹/³

Evaluate this exact answer and compare it with the value you found in part (b).

[Technology Exercise] Roots

Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.

a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = x√(1 − x²)

Derivatives in Differential Form

In Exercises 17–28, find dy.

2y³/² + xy − x = 0

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sec(x² − 1)

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = sin(5√x)

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = 3 csc(1 − 2√x)