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

Derivatives as Functions

Calculus

2. Intro to Derivatives

Derivatives as Functions

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Multiple Choice

Given that the second derivative of a function is $f^{″} (x) = 2 x - \cos (x)$ , which of the following is a possible form for the original function $f (x)$ ?

f (x) = x^{3} + \sin (x) + C_{1}

f (x) = x^{2} + \cos (x) + C_{1} x + C_{2}

f (x) = x^{3} - \sin (x) + C_{1} x + C_{2}

f (x) = x^{2} - \sin (x) + C_{1}

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Verified step by step guidance

Step 1: Understand the problem. We are given the second derivative of a function, f''(x) = 2x - cos(x), and need to determine which of the provided options could represent the original function f(x). This involves integrating the second derivative twice to recover f(x).

Step 2: Perform the first integration. Integrate f''(x) = 2x - cos(x) with respect to x to find the first derivative f'(x). The integral of 2x is x^2, and the integral of -cos(x) is -sin(x). Add an arbitrary constant C₁ to account for the constant of integration. Thus, f'(x) = x^2 - sin(x) + C₁.

Step 3: Perform the second integration. Integrate f'(x) = x^2 - sin(x) + C₁ with respect to x to find f(x). The integral of x^2 is (1/3)x^3, the integral of -sin(x) is cos(x), and the integral of C₁ is C₁x. Add another arbitrary constant C₂ to account for the constant of integration. Thus, f(x) = (1/3)x^3 + cos(x) + C₁x + C₂.

Step 4: Compare the derived f(x) with the given options. The derived f(x) = (1/3)x^3 + cos(x) + C₁x + C₂ matches the structure of one of the provided options, but note that the coefficients may differ slightly due to simplifications or adjustments.

Step 5: Verify the correct answer. Based on the comparison, the correct answer is f(x) = x^3 - sin(x) + C₁x + C₂, as it aligns with the integration steps and the given second derivative.