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CLEAR Calculus

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Lab 2: Vector-valued functions

  1. An object is moving through space with position vector given by \({\bf r}(t)=\langle \cos 2t, \sin 2t, t\rangle\) where time \(t\) is measured in seconds since the start of motion and the \(x\), \(y\), and \(z\) coordinates are measured in meters.

      1. Describe the path traveled by the particle.

      2. Where is the particle 1 second after the start of motion? Where is it \(\frac{1}{10}\) second after that?

      3. Use your answers to Part (a) and the distance formula to approximate the speed of the particle at time \(t=1\). Is this an underestimate or overestimate? Explain how you know.

      4. For a vector-valued function \({\bf r}(t)=\langle x(t),y(t),z(t)\rangle\), the derivative is defined as the vector-valued function \({\bf r}'(t)=\langle x'(t),y'(t),z'(t)\rangle\). Compute both \({\bf r}'(1)\) and \(\|{\bf r}'(1)\|\) and explain what each one means.

  2. Consider the planar curve defined by \({\bf p}(t)=\langle\frac{\ln t}{1-t}, t^{1/(1-t)}\rangle\) for values of \(t>0\). Approximate the location of the hole in the curve from the undefined values when \(t=1\). How accurate is your approximation? Explain how you know.

  3. A particle starts at \((0,0)\) and moves in the plane with velocity \({\bf q}'(t)=\langle t,\sin t\rangle\).

    1. Where is the particle at time \(t=2\)? That is, what is \({\bf q}(2)\)?

       

    2. Find a formula for the speed of the particle at time \(t\) and sketch its graph. (Remember speed is a scalar.) Is the particle speeding up or slowing down at time \(t=2\)?

       

    3. Compute \({\bf q}'(2)\cdot{\bf q}''(2)\). How does this relate to your answer to Part (b)?

       

    4. Find both an underestimate and overestimate for how far the particle traveled from \(t=0\) to \(t=2\). (Note this is asking for the distance traveled along its path, not just the net displacement.)


  4. A central force is one which acts on an object in a direction parallel to the position vector of that object.

    1. Give at least two examples of physical situations in which a central force is involved. In your examples does the force vector point away from or towards the origin? Can you think of examples in which the force vector points in the opposite direction?

    2. Prove that the orbit of the earth lies entirely within a plane containing the sun. Hint: differentiate \({\bf r}\times{\bf r}'\) with respect to time, where \({\bf r}\) is the earth's position vector and \({\bf r}'\) is its velocity.