2011 Calculus BC free response #1 (b & c) | AP Calculus BC | Khan Academy

TL;DR

The video explains how to find the slope of the tangent line and the position of a particle at time t equals 3.

Transcript

Part b-- find the slope of the tangent line to the path of the particle at time t is equal to 3. So the slope of the tangent line is just going to be equal to the rate of change of y with respect to x at that point. And this is the same thing as dy/dt over dx/dt. And dealing with differentials is a little bit strange, especially when you want to de... Read More

Key Insights

• ⏳ The slope of the tangent line to the particle's path at time t equals 3 can be determined by finding the ratio of dy/dt to dx/dt.
• 🍞 Multiplying both the numerator and denominator by dt/dt cancels out the differentials and simplifies the calculation of the slope.
• ⏳ Finding dy/dt and dx/dt at time t equals 3 allows for the calculation of the slope of the tangent line.
• 🧘 The position of the particle at time t equals 3 can be found by evaluating the functions x(t) and y(t) at t equals 3.
• 🧘 The antiderivative of a function is used to find the position of the particle by integrating the given functions.
• ❓ The constant of integration in the antiderivative can be determined using an initial condition provided in the problem.

Questions & Answers

Q: How is the slope of the tangent line related to the rate of change of y with respect to x?

The slope of the tangent line is equal to the ratio of dy/dt to dx/dt, as explained in the video. It represents the rate of change of y with respect to x at a specific point in time.

Q: Why does multiplying both the numerator and denominator by dt/dt cancel out the differentials?

Multiplying both the numerator and denominator by dt/dt is an algebraic manipulation to get rid of the differentials. When viewed as small changes in y, x, and t, the cancellation occurs because dt/dt represents the same small change in t for both numerator and denominator.

Q: How do you find dy/dt at time t equals 3?

Dy/dt at time t equals 3 is equal to the sine of 3 squared, or sine of 9. This value is given in the video and can be calculated as approximately 0.0317.

Q: What is the position of the particle at time t equals 3?

The position of the particle at time t equals 3 is (21, -3.226). This is obtained by evaluating the functions x(t) and y(t) at t equals 3, as shown in the video.

Summary & Key Takeaways

• The slope of the tangent line to the path of the particle at time t equals 3 can be found by taking the derivative of y with respect to x.

• To find the slope, the video demonstrates multiplying both the numerator and the denominator by dt/dt, canceling them out, and obtaining the ratio of dy/dt to dx/dt.

• The video then explains how to find the position of the particle at time t equals 3 by integrating the given functions and evaluating them at that time.