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

Introduction to Functions

Calculus

0. Functions

Introduction to Functions

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Multiple Choice

Let $r (t)$ be the vector-valued function defined by $r (t) = (3 t, \frac{1}{t + 1}, e^{- t})$ . Evaluate the definite integral from $t = 0$ to $t = 1$ of $r (t) d t$ .

(\frac{3}{2}, \ln (2), 1 - \frac{1}{e})

(3, \ln (2), e^{- 1})

(3, 1, \frac{1}{e})

(1, \frac{1}{2}, e - 1)

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Verified step by step guidance

Step 1: Understand the problem. You are tasked with evaluating the definite integral of the vector-valued function r(t) = (3t, 1/(t+1), e^{-t}) from t = 0 to t = 1. This involves integrating each component of the vector function separately over the given interval.

Step 2: Break down the vector-valued function into its components: r(t) = (3t, 1/(t+1), e^{-t}). The first component is 3t, the second component is 1/(t+1), and the third component is e^{-t}. Each of these will be integrated individually.

Step 3: Set up the definite integrals for each component: For the first component: $ \int_{0}^{1} 3t \, dt $ For the second component: $ \int_{0}^{1} \frac{1}{t+1} \, dt $ For the third component: $ \int_{0}^{1} e^{-t} \, dt $.

Step 4: Solve each integral step by step: - For $ \int_{0}^{1} 3t \, dt $, use the power rule for integration: $ \int t^n \, dt = \frac{t^{n+1}}{n+1} $. - For $ \int_{0}^{1} \frac{1}{t+1} \, dt $, recognize this as the natural logarithm function: $ \int \frac{1}{x} \, dx = \ln|x| $. - For $ \int_{0}^{1} e^{-t} \, dt $, use the exponential rule for integration: $ \int e^{ax} \, dx = \frac{1}{a} e^{ax} $.

Step 5: Evaluate each integral over the interval [0, 1]: - For $ \int_{0}^{1} 3t \, dt $, substitute the limits of integration after applying the power rule. - For $ \int_{0}^{1} \frac{1}{t+1} \, dt $, substitute the limits of integration into the natural logarithm function. - For $ \int_{0}^{1} e^{-t} \, dt $, substitute the limits of integration into the exponential function. Combine the results into a vector to represent the final answer.