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

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3:23 minutes

Problem 3.3.57

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

Verified step by step guidance

To ensure the function g(x) is differentiable at x = 0, it must be continuous at x = 0. This means the left-hand limit as x approaches 0 from the negative side must equal the right-hand limit as x approaches 0 from the positive side.

Calculate the left-hand limit of g(x) as x approaches 0 from the negative side: lim(x → 0⁻) ax = a * 0 = 0.

Calculate the right-hand limit of g(x) as x approaches 0 from the positive side: lim(x → 0⁺) (x² - 3x) = 0² - 3*0 = 0.

Since both limits are equal, the function is continuous at x = 0. Now, ensure the derivatives from both sides are equal at x = 0 for differentiability.

Find the derivative of g(x) for x < 0, which is g'(x) = a. Find the derivative of g(x) for x ≥ 0, which is g'(x) = 2x - 3. Set these equal at x = 0: a = 2*0 - 3 = -3. Therefore, a must be -3 for differentiability.

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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. This means that the function must be continuous at that point, and the left-hand and right-hand derivatives must be equal. In this case, we need to ensure that the function g(x) is differentiable at x = 0, where the definition of the function changes.

Continuity

A function is continuous at a point if the limit of the function as it approaches that point from both sides equals the function's value at that point. For g(x) to be differentiable at x = 0, it must first be continuous there. This requires that the left-hand limit (as x approaches 0 from the left) equals the right-hand limit (as x approaches 0 from the right) and that both equal g(0).

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, g(x) has two pieces: ax for x < 0 and x² - 3x for x ≥ 0. To analyze differentiability, we must examine the behavior of both pieces at the point where they meet (x = 0) and ensure that they connect smoothly without any jumps or breaks.

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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

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

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).

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

Textbook Question

b. Show that