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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Multiple Choice

Which of the following functions is a solution to the differential equation $\frac{d y}{d x} = e^{2 x} + 2 y$ ?

y = \frac{1}{3} e^{2 x}

y = \frac{1}{3} e^{2 x} + C e^{2 x}

y = e^{2 x}

y = e^{2 x} + C e^{2 x}

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Verified step by step guidance

Step 1: Begin by analyzing the given differential equation $ \frac{dy}{dx} = e^{2x} + 2y $. This equation is a first-order linear differential equation, and we aim to verify which of the provided functions satisfies it.

Step 2: Substitute each candidate function into the differential equation. For example, if $ y = \frac{1}{3} e^{2x} $, compute $ \frac{dy}{dx} $ and check if $ \frac{dy}{dx} = e^{2x} + 2y $ holds true.

Step 3: For the function $ y = \frac{1}{3} e^{2x} + Ce^{2x} $, compute $ \frac{dy}{dx} $ using the derivative rules for exponential functions. Substitute $ \frac{dy}{dx} $ and $ y $ into the differential equation to verify if it satisfies $ \frac{dy}{dx} = e^{2x} + 2y $.

Step 4: Repeat the substitution process for the other candidate functions, $ y = e^{2x} $ and $ y = e^{2x} + Ce^{2x} $, to determine if they satisfy the differential equation.

Step 5: Compare the results of the substitutions for all candidate functions. The correct solution will be the function that satisfies the differential equation $ \frac{dy}{dx} = e^{2x} + 2y $ for all values of $ x $.