Based on the graph f(x)f\left(x\right)f(x), describe all points where the derivative f′(x)f^{\prime}\left(x\right)f′(x)would have a jump.

x=−1x=-1x=−1 and x=0.5x=0.5x=0.5

Based on the graph f ( x ) f\left(x\right) f ( x ) , describe all points where the derivative f ′ ( x ) f^{\prime}\left(x\right) f ′ ( x ) would have a jump.

So in this problem, we're told based on the graph, f of x, describe all the points where the derivative f prime of x would have a jump. So this would, in essence, mean there's a discontinuity in the derivative graph, and recall that this happens whenever the function is discontinuous at a point or has a sharp turn. Now looking at this function here, I can trace this whole function without picking up the pen. And because of that, that means that this is continuous everywhere, but it's not necessarily going to have a derivative that exists everywhere because we need to watch out for sharp turns or corners as well. Well, I can see that there's a sharp corner that appears right about here, and I can see that there's a sharp corner that appears right there. So what that means is at x equals negative one and at x equals positive 0.5 or 1 half, there is going to be a jump in our derivative graph, and that is going to be the solution to this problem. So I hope you found this helpful, and let's move on.

Based on the graph f(x)f\left(x\right)f(x), describe all points where the derivative f′$$(x)f^{\prime}$$\left(x\right)f′(x) would have a jump.

x=−1x=-1x=−1 and x=0.5x=0.5x=0.5

x=−1.5x=-1.5x=−1.5, x=−1x=-1x=−1, and x=0.5x=0.5x=0.5

Derivative f′$$(x)f^{\prime}$$\left(x\right)f′(x) has no jumps

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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 36m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 16m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
30m

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

Basic Graphing of the Derivative

Business Calculus

2. Intro to Derivatives

Basic Graphing of the Derivative

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Multiple Choice

Based on the graph $f$\left$(x$\right$)$ , describe all points where the derivative $f^{$\prime$}$\left$(x$\right$)$ would have a jump.

$x=-1.5$ , $x=-1$ , and $x=0.5$

$x=-1$ and $x=0.5$

$x=-1.5$

Derivative $f^{$\prime$}$\left$(x$\right$)$ has no jumps

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Verified step by step guidance

Step 1: Observe the graph of f(x) and identify the points where the slope of the graph changes abruptly. These points are potential locations where the derivative f'(x) might have a jump.

Step 2: At x = -1.5, the graph of f(x) transitions smoothly without any abrupt change in slope. Therefore, the derivative f'(x) does not have a jump at this point.

Step 3: At x = -1, the graph of f(x) shows a sharp corner where the slope changes abruptly. This indicates that the derivative f'(x) has a jump at x = -1.

Step 4: At x = 0.5, the graph of f(x) again shows a sharp corner with an abrupt change in slope. This confirms that the derivative f'(x) has a jump at x = 0.5.

Step 5: Summarize the findings: The derivative f'(x) has jumps at x = -1 and x = 0.5, but not at x = -1.5.