Determine if the graph of the function f ( x ) f\left(x\right) f ( x ) is continuous and/or differentiable at x = 2 x=2 x = 2 .

Let's try solving this problem. So here we're asked to determine if the graph of the function f of x is continuous and or differentiable at x equals 2. So the point x equals 2 is going to be right there, and the way that we can figure this out is by first figuring out if it's continuous and then figuring out if it's differentiable. Because recall that it's possible for a graph to be continuous but not differentiable, but it's not possible for a graph to be differentiable but not continuous at a certain point. So let's see what happens. Well, to figure out if this is continuous, I'm gonna trace this with my pen. And as I can see, I don't actually have to pick up my pen at any point. I can trace this all in one curve, so that means that it is continuous. But the question is, is it now differentiable? Well, if it's continuous, that's one of the criteria for being differentiable. But to figure out if this is differentiable, it can't have any sharp corners or turns. Well, looking at this, I don't see any sharp corners or turns. It looks like it's actually a smooth turn right there. So because of this, we could say it's continuous and differentiable at x equals 2. So we could say that this is continuous and differentiable, and that would be the solution to this problem. So that is how you can solve these types of problems. Hope you found this video helpful.

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0. Functions4h 53m

Introduction to Functions
16m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent Rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
34m

Exponential & Logarithmic Equations
35m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation2h 18m

Basic Rules of Differentiation
43m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

The Chain Rule
27m

4. Derivatives of Exponential & Logarithmic Functions1h 16m

Derivatives of Exponential & Logarithmic Functions
1h 16m

5. Applications of Derivatives2h 19m

Implicit Differentiation
27m

Logarithmic Differentiation
17m

Related Rates
59m

Differentials
21m

Linearization
13m

6. Graphical Applications of Derivatives6h 0m

Intro to Extrema
21m

The First Derivative Test
1h 20m

Concavity
1h 0m

The Second Derivative Test
22m

Curve Sketching
44m

Finding Global Extrema
29m

Applied Optimization
1h 41m

7. Antiderivatives & Indefinite Integrals48m

Antiderivatives
17m

Indefinite Integrals
31m

8. Definite Integrals4h 30m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 16m

Introduction to Definite Integrals
20m

Fundamental Theorem of Calculus
25m

Average Value of a Function
17m

Substitution for Indefinite Integrals
55m

Substitution for Definite Integrals
32m

9. Graphical Applications of Integrals1h 43m

Area Between Curves
1h 19m

Introduction to Volume & Disk Method
23m

10. Integrals of Inverse, Exponential, & Logarithmic Functions21m

Integrals of Exponential Functions
14m

Integrals Involving Logarithmic Functions
7m

11. Techniques of Integration2h 7m

Integration by Parts
1h 37m

Improper Integrals
29m

12. Trigonometric Functions6h 54m

Angles
29m

Trigonometric Functions on Right Triangles
1h 8m

Solving Right Triangles
23m

Trigonometric Functions on the Unit Circle
1h 19m

Graphs of Sine & Cosine
46m

Graphs of Other Trigonometric Functions
32m

Trigonometric Identities
52m

Derivatives of Trig Functions
42m

Integrals of Basic Trig Functions
28m

Integrals of Other Trig Functions
10m

13: Intro to Differential Equations2h 23m

Basics of Differential Equations
43m

Slope Fields
19m

Separable Differential Equations
1h 21m

14. Sequences & Series2h 8m

Sequences
1h 12m

Review of Factorials
11m

Series
44m

15. Power Series2h 19m

Power Series & Taylor Series
2h 19m

16. Probability & Calculus45m

Continuous Probability Models
45m

2. Intro to Derivatives

Differentiability

Business Calculus

2. Intro to Derivatives

Differentiability

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Determine if the graph of the function $f$\left$(x$\right$)$ is continuous and/or differentiable at $x=2$ .

Continuous and non-differentiable

Continuous and differentiable

Discontinuous and non-differentiable

Discontinuous and differentiable

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Step 1: Analyze the graph of the function f(x) at x=2. Observe whether there is any break, jump, or hole in the graph at x=2. If the graph is unbroken and continuous at x=2, the function is continuous at this point.

Step 2: Check the slope of the graph at x=2. If the graph has a smooth curve without any sharp corners or vertical tangents at x=2, the function is differentiable at this point.

Step 3: Recall that for a function to be differentiable at a point, it must also be continuous at that point. Therefore, if the graph is continuous and smooth at x=2, the function is both continuous and differentiable.

Step 4: Based on the graph provided, confirm that the function f(x) does not have any discontinuities (breaks, jumps, or holes) at x=2 and that the slope of the graph changes smoothly without sharp corners or vertical tangents.

Step 5: Conclude that the function f(x) is continuous and differentiable at x=2 based on the observations from the graph.