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

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

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Multiple Choice

Which of the following definite integrals is equal to $lim_{n \to \infty} \sum \frac{10}{n} (1 + \frac{5 k}{n})^{2} (k = 1 n)$ ?

\int_{0}^{5} 10 x^{2} d x

\int_{1}^{6} 50 x^{2} d x

\int_{1}^{6} 10 x^{2} d x

\int_{0}^{5} 50 x^{2} d x

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Verified step by step guidance

Step 1: Recognize that the given limit represents a Riemann sum. A Riemann sum is a way to approximate the value of a definite integral by summing up areas of rectangles under a curve. The general form of a Riemann sum is lim_{n \to \infty} \sum_{k=1}^{n} f(x_k) \Delta x, where \Delta x is the width of each subinterval and f(x_k) is the function evaluated at a sample point in each subinterval.

Step 2: Identify the components of the given Riemann sum. Here, \Delta x = \frac{10}{n}, which represents the width of each subinterval. The term \left(1 + \frac{5k}{n}\right)^2 corresponds to the function being integrated, and the interval of integration can be inferred from the range of x values covered by the sum.

Step 3: Determine the interval of integration. The term \left(1 + \frac{5k}{n}\right) suggests that the x-values range from 1 (when k=1) to 6 (when k=n). This means the definite integral will be evaluated over the interval [1, 6].

Step 4: Write the function being integrated. The given Riemann sum includes the term \left(1 + \frac{5k}{n}\right)^2, which corresponds to the function f(x) = 50x^2 when multiplied by the constant factor 50 (from the term 5n). Thus, the integral represents \int_{1}^{6} 50x^2 \, dx.

Step 5: Conclude that the correct answer is \int_{1}^{6} 50x^2 \, dx, as this matches the function and interval derived from the Riemann sum.