Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

2. Intro to Derivatives

Differentiability

Calculus

2. Intro to Derivatives

Differentiability

Previous problemNext problem

3:55 minutes

Problem 3.2.60a

Textbook Question

a. Let f(x) be a function satisfying |f(x)| ≤ x² for −1 ≤ x ≤ 1. Show that f is differentiable at x = 0 and find f′(0).

Verified step by step guidance

To show that f is differentiable at x = 0, we need to check if the limit \( \lim_{h \to 0} \frac{f(h) - f(0)}{h} \) exists. Since |f(x)| ≤ x², we know that |f(h)| ≤ h² for small values of h.

Evaluate the expression \( \frac{f(h) - f(0)}{h} \). Since f(0) is bounded by 0², we have |f(0)| ≤ 0, which implies f(0) = 0.

Substitute f(0) = 0 into the limit expression: \( \lim_{h \to 0} \frac{f(h)}{h} \). Given |f(h)| ≤ h², we have \( -h^2 \/ ≤ f(h) \/ ≤ h^2 \).

Divide the inequality by h (assuming h ≠ 0): \( -h \/ ≤ \frac{f(h)}{h} \/ ≤ h \). As h approaches 0, both -h and h approach 0.

By the Squeeze Theorem, \( \lim_{h \to 0} \frac{f(h)}{h} = 0 \). Therefore, f is differentiable at x = 0 and f′(0) = 0.

Verified video answer for a similar problem:

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

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Differentiability

A function is differentiable at a point if it has a defined derivative at that point, which means the limit of the difference quotient exists. For a function f(x) to be differentiable at x = 0, the limit of (f(x) - f(0)) / (x - 0) as x approaches 0 must exist. Differentiability implies continuity, so if a function is differentiable at a point, it must also be continuous there.

Limit Definition of Derivative

The derivative of a function at a point can be defined using the limit: f'(a) = lim (h -> 0) [f(a + h) - f(a)] / h. In this case, to find f'(0), we need to evaluate the limit as h approaches 0 of [f(h) - f(0)] / h. This definition is crucial for determining the slope of the tangent line to the function at that point.

Bounded Functions

The condition |f(x)| ≤ x² for −1 ≤ x ≤ 1 indicates that the function f(x) is bounded by the parabola x² within the specified interval. This constraint helps in analyzing the behavior of f(x) as x approaches 0, particularly in ensuring that f(x) approaches 0 faster than x does, which is essential for proving differentiability at that point.

Watch next

Master Determining Differentiability Graphically with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

c. neither continuous nor differentiable?

Give reasons for your answers.

Textbook Question

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

c. neither continuous nor differentiable?

Give reasons for your answers.

Textbook Question

Differentiability and Continuity on an Interval

Each figure in Exercises 45–50 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable?

Give reasons for your answers.

Textbook Question

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.

Textbook Question

Find the value of a that makes the following function differentiable for all x-values.

g(x) = { ax, if x < 0

x² − 3x, if x ≥ 0

Textbook Question

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

g(x) = { x²/³, x ≥ 0

x¹/³, x < 0

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