Math 344: Calculus III

Midterm Exam

Review Guide

1. Find an equation of the plane through the points \(P(1,2,3)\), \(Q(4,0,-1)\), and \(R(2,-4,-2)\).

2. Find the distance between the point and the given plane. \[\begin{cases} P(1,-3,2) \\ \Pi:\ \ 3x + 2y + 6z = 5 \end{cases}\]

3. Reduce the equation to one of the standard forms and classify the surface. \[x^2 - y^2 - z^2 - 4x - 2z +3 = 0\]

4. You must be able to match the graph of a surface in \(\mathbb{R}^3\) to its equation. (This will be a multiple choice question.)

5. Consider the vector function \[\mathbf{r}(t) = \dfrac{t^2-1}{t-1}\,\mathbf{i} + \sqrt{t+8}\,\mathbf{j} + \dfrac{\sin \pi t}{\ln t}\,\mathbf{k}.\]

	a.)	What is the domain of \(\mathbf{r}\)?
	b.)	Compute \(\displaystyle \lim_{t\to 1} \mathbf{r}(t)\), provided it exists.
	c.)	Compute \(\dot{\mathbf{r}}(t)\), provided it exists.

6. Find a parametrization of the curve of intersection of the cylinder \(x^2 + y^2 = 4\) and the surface \(z = xy\).

(Recall that "parametrization" is another name for a vector function whose terminal points trace out the space curve.)

7. Let \(\mathbf{u}\) and \(\mathbf{v}\) be vector functions in \(\mathbb{R}^3\). Prove the product rule for the dot product: \[ \frac{d}{dt}\Big[ \mathbf{u}(t)\cdot\mathbf{v}(t)\Big] = \dot{\mathbf{u}}(t)\cdot\mathbf{v}(t) + \mathbf{u}(t)\cdot\dot{\mathbf{v}}(t).\]

Hint. Write both \(\mathbf{u}\) and \(\mathbf{v}\) in coordinates.

8. Consider the vector function \(\mathbf{r}(t) = \left\langle \arctan(t), 2e^{2t}, 8te^t\right\rangle.\) Find the unit tangent vector \(\mathbf{T}(0)\).

9. Find \(\mathbf{r}(t)\) if \(\dot{\mathbf{r}}(t) = \left\langle t, e^t, te^t \right\rangle\) and \(\mathbf{r}(0) = \mathbf{i} + \mathbf{j} + \mathbf{k}\).

10. Reparametrize the plane curve \[\mathbf{r}(t) = \left(\dfrac{2}{t^2+1} - 1\right)\,\mathbf{i} + \dfrac{2t}{t^2+1}\,\mathbf{j}\] with respect to arc length from the point \((1,0)\) in the direction of increasing \(t\). Express the reparametrization in simplest form. What can you deduce about the curve?

11. Compute the curvature \(\kappa\) of the twisted cubic \(\mathbf{r}(t) = \left\langle t, t^2, t^3 \right\rangle\) at the point \(P(1,1,1)\).

12. At what point on the curve \(x = t^3\), \(y = 3t\), \(z = t^4\) is the normal plane parallel to the plane \(6x + 6y - 8z = 1\)?

Hint. The normal plane has normal vector \(\mathbf{T}\): it is spanned by the normal and binormal vectors, \(\mathbf{N}\) and \(\mathbf{B}\).

13. Show that the osculating plane at every point on the curve \(\mathbf{r}(t) = \left\langle t+2, 1 - t, \tfrac{1}{2}t^2 \right\rangle\) is the same plane. What can you conclude about the curve?

14. Let \(C\) be a smooth space curve with unit tangent vector field \(\mathbf{T}\). Prove that \(\mathbf{T} \perp \dot{\mathbf{T}}\) for all \(t\) in the domain of \(\mathbf{T}\).

15. Prove that the curvature of a circle of radius \(a\) is constant, \(\kappa = 1/a \).

16. Show that the limit does not exist. \[\lim_{(x,y)\to (0,0)}\ \dfrac{x^4 - 30y^2}{x^2 + 15y^2}\]

17. Determine the signs of the partial derivatives for the function \(f\) whose graph is show, at the point \((x_0,y_0)\).


18. Compute the second directional derivative \(D_{\mathbf{v}}^2 f(x,y) = D_{\mathbf{v}}[D_{\mathbf{v}} f(x,y)]\) for \(f(x,y) = x^3 + 5x^2y + y^3\) in the direction of the vector \(\mathbf{v} = \langle 3,4 \rangle\) and evaluate it at the point \((3,2)\).

19. Consider the function \(f(x,y) = xy\) at the point \(P(6,5)\), and consider the level curve \(f(x,y) = 30\). Find an equation of the tangent line to the level curve at \(P\), and compute the gradient of \(f\) at \(P\). Then show that the gradient vector is perpendicular to the level curve at \(P\).

20. Prove the theorem: Suppose \(f\) is a differentiable function of at least 2 variables. The maximum value of the directional derivative \(D_{\mathbf{u}}f(\mathbf{x})\) is \(\Vert\nabla f(\mathbf{x})\Vert\) and it occurs when \(\mathbf{u}\) has the same direction as the gradient vector \(\nabla f(\mathbf{x})\).

21. Find an equation of the osculating circle to the curve \(y = 4x^2 - x^4\) at \(x = 2\).

22. The involute of a space curve \(\mathbf{r}(t)\) is the space curve defined by \(\ \mathcal{I}(t) = \mathbf{r}(t) - s(t)\mathbf{T}(t)\), where \(s(t)\) is the arc length function. Show that the involute of the helix, \(\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle\) lies in the \(xy\)-plane.

23. Use a \(\delta\)-\(\varepsilon\) argument to prove that the limit does exist. \[ \lim_{(x,y) \to (0,0)}\ \frac{5xy^2}{x^2 + y^2} \]

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