Derivatives in Differential Form


In Exercises 17–28, find dy.


2y³/² + xy − x = 0

[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.

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)

Approximation Error

In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find

a. the change Δf = f(x₀ + dx) − f(x₀);

b. the value of the estimate df = fʹ(x₀) dx; and

c. the approximation error |Δf − df|.

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f(x) = x² + 2x, x₀ = 1, dx = 0.1