Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.2.58

Chapter 3, Problem 3.2.58

Derivative of multiples Does knowing that a function g(t) is differentiable at t = 7 tell you anything about the differentiability of the function 3g at t = 7? Give reasons for your answer.

Verified step by step guidance

To determine the differentiability of the function 3g at t = 7, we need to understand the concept of differentiability for a function. A function is differentiable at a point if it has a derivative at that point.

Given that g(t) is differentiable at t = 7, it means that the derivative g'(7) exists.

The function 3g(t) is a scalar multiple of g(t). The derivative of a scalar multiple of a function is the scalar multiplied by the derivative of the function. This is a basic rule of differentiation.

Therefore, the derivative of 3g(t) with respect to t is 3 times the derivative of g(t). Mathematically, this can be expressed as (3g)'(t) = 3 * g'(t).

Since g'(7) exists, it follows that (3g)'(7) = 3 * g'(7) also exists. Thus, the function 3g is differentiable at t = 7.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Differentiability

Differentiability of a function at a point means that the function has a defined derivative at that point. This implies the function is smooth and continuous at that point, without any sharp corners or discontinuities. For a function g(t) to be differentiable at t = 7, it must have a well-defined tangent line at that point.

Derivative of a Constant Multiple

The derivative of a constant multiple of a function is the constant multiplied by the derivative of the function. Mathematically, if g(t) is differentiable, then the derivative of 3g(t) is 3 times the derivative of g(t). This rule ensures that scaling a function by a constant does not affect its differentiability.

Linearity of Differentiation

Differentiation is a linear operation, meaning it respects addition and scalar multiplication. This property implies that if g(t) is differentiable at a point, then any linear transformation of g(t), such as 3g(t), is also differentiable at that point. Thus, knowing g(t) is differentiable at t = 7 ensures 3g(t) is differentiable at t = 7.

Recommended video:

07:17

Linearization

Related Practice

Textbook Question

Find all points on the curve y = tan x, −π/2 < x < π/2, where the tangent line is parallel to the line y = 2x. Sketch the curve and tangent lines together, labeling each with its equation.

255

views

Textbook Question

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

y = x³/3 + x²/2 + x/4

265

views

Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = cos(x²)

173

views

Textbook Question

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|.

f(x) = x⁴, x₀ = 1, dx = 0.1

181

views

Textbook Question

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

g(x) = { 2x − x³ − 1, x ≥ 0

x − (1 / (x + 1)), x < 0

306

views

Textbook Question

Slopes and Tangent Lines

In Exercises 1–4, use the grid and a straight edge to make a rough estimate of the slope of the curve (in y-units per x-unit) at the points P₁ and P₂.

256

views