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

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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Multiple Choice

Find the exact length of the curve $y = x - x^{2} + {\sin - 1}^{x}$ from $x = 0$ to $x = 1$ .

\int_{0}^{1} \sqrt{1 + {(1 - 2 x)}^{2}} d x

\int_{0}^{1} \sqrt{1 + {(1 - 2 x + \frac{1}{\sqrt{1 - x^{2}}})}^{2}} d x

\int_{0}^{1} \sqrt{1 + {(1 + 2 x + \frac{1}{\sqrt{1 - x^{2}}})}^{2}} d x

\int_{0}^{1} \sqrt{1 + {(1 - 2 x - \frac{1}{\sqrt{1 - x^{2}}})}^{2}} d x

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Verified step by step guidance

Step 1: Recall the formula for the arc length of a curve y = f(x) from x = a to x = b. The formula is given by: $ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx $. Here, we need to compute the derivative $ \frac{dy}{dx} $ for the given function y = x - x^2 + \sin^{-1} x.

Step 2: Differentiate y = x - x^2 + \sin^{-1} x with respect to x. Using the derivative rules, $ \frac{d}{dx}(x) = 1 $, $ \frac{d}{dx}(x^2) = 2x $, and $ \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}} $, we find $ \frac{dy}{dx} = 1 - 2x + \frac{1}{\sqrt{1 - x^2}} $.

Step 3: Substitute $ \frac{dy}{dx} $ into the arc length formula. This gives $ L = \int_0^1 \sqrt{1 + \left(1 - 2x + \frac{1}{\sqrt{1 - x^2}}\right)^2} \, dx $.

Step 4: Simplify the integrand if possible. Expand $ \left(1 - 2x + \frac{1}{\sqrt{1 - x^2}}\right)^2 $ and combine terms under the square root. This step involves algebraic manipulation, but the integral remains in its current form.

Step 5: Recognize that the integral $ \int_0^1 \sqrt{1 + \left(1 - 2x + \frac{1}{\sqrt{1 - x^2}}\right)^2} \, dx $ represents the exact length of the curve. Evaluate the integral using numerical methods or advanced techniques if required, but the setup is complete.