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

0. Functions

Piecewise Functions

Calculus

0. Functions

Piecewise Functions

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Multiple Choice

Which of the following correctly describes the piecewise function graphed below, where the graph consists of a line with slope $2$ for $x < 1$ and a constant value $3$ for $x \geq 1$ ?

f (x) = 2 x

for

x < 1

;

f (x) = 3

for

x \geq 1

f (x) = 3

for

x < 1

;

f (x) = 2 x

for

x \geq 1

f (x) = 2 x

for

x < 1

;

f (x) = x + 3

for

x \geq 1

f (x) = 2 x + 1

for

x < 1

;

f (x) = 3

for

x \geq 1

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Verified step by step guidance

Step 1: Understand the problem. The question asks us to identify the correct piecewise function that describes the given graph. The graph consists of two parts: a line with slope 2 for x < 1, and a constant value of 3 for x ≥ 1.

Step 2: Recall the general form of a piecewise function. A piecewise function is defined as different expressions for different intervals of the domain. For example, f(x) = {expression1 for interval1; expression2 for interval2}.

Step 3: Analyze the first part of the graph. For x < 1, the graph is described as a line with slope 2. The equation of a line is generally written as y = mx + b, where m is the slope. Since the slope is 2, the equation for this part is f(x) = 2x.

Step 4: Analyze the second part of the graph. For x ≥ 1, the graph is described as a constant value of 3. A constant function is written as f(x) = c, where c is the constant value. Therefore, for this part, f(x) = 3.

Step 5: Combine the two parts into a piecewise function. Based on the analysis, the piecewise function is f(x) = 2x for x < 1; f(x) = 3 for x ≥ 1. This matches the first option provided in the problem.