Table of contents
- 0. Functions7h 52m
- Introduction to Functions16m
- Piecewise Functions10m
- Properties of Functions9m
- Common Functions1h 8m
- Transformations5m
- Combining Functions27m
- Exponent rules32m
- Exponential Functions28m
- Logarithmic Functions24m
- Properties of Logarithms34m
- Exponential & Logarithmic Equations35m
- Introduction to Trigonometric Functions38m
- Graphs of Trigonometric Functions44m
- Trigonometric Identities47m
- Inverse Trigonometric Functions48m
- 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 Integrals3h 25m
4. Applications of Derivatives
Motion Analysis
Problem 49b
Textbook Question
The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.
b. Find the velocity function for the marble.
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1
Identify the position function given in the problem: s(t) = 100t / (t + 1).
To find the velocity function, recall that velocity is the derivative of the position function with respect to time, v(t) = ds/dt.
Apply the quotient rule for differentiation, which states that if you have a function s(t) = u(t)/v(t), then ds/dt = (u'v - uv')/v^2, where u = 100t and v = t + 1.
Calculate the derivatives u' and v': u' = 100 and v' = 1.
Substitute u, u', v, and v' into the quotient rule formula to find the velocity function v(t).
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