Q: Find the length of the curve $$\mathbf{r}(t) = \langle \cos t, \sin t, \ln \cos t \rangle$$ for $$0 \leq t \leq \frac{\pi}{4}.$$

$$\ln \left| \sec \frac{\pi}{4} + \tan \frac{\pi}{4} \right|$$

Q: Find the angle $$\theta$$ between the planes $$x + y - z = 1$$ and $$2x - 3y + 4z = 5. $$Which formula should you use?

$$\cos \theta = \frac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|}$$

Question 1

Given the points $$(3, -1, 1)$$, $$(4, 0, 2)$$, and $$(6, 3, 1)$$, which of the following is an equation of the plane containing these points?

Accepted Answer

$$2x - 3y + z = 8$$

Question 2

A particle moves in space with position $$\mathbf{r}(t) = \langle t^2$, e^t, t e^t \rangle. $$What is the speed of the particle at time $$t=0$$?

Accepted Answer

$$1$$

Question 3

Find the point where the line $$\mathbf{r}(t) = \langle 2 - t, 1 + 3t, 4t \rangle$$ intersects the plane $$2x - y + z = 2.$$

Accepted Answer

Question 4

Let $$f(x, y) = \ln(9 - x^2$ - 9y^2$). $$What is the domain of $$f$$?

Accepted Answer

$$x^2$$ + $$9y^2$$ \u003c 9$$

Question 5

Which of the following statements is true about the planes $$x + y - z = 1$$ and $$2x - 3y + 4z = 5$$?

Accepted Answer

They are neither parallel nor perpendicular.

Question 6

Let $$f(x, y) = 3xy^2$ - 2y + 5x^2$y^2. $$Find the directional derivative of $$f at$$ $$(-1, 2) in $$the direction of $$\mathbf{u} = \langle -1, 2 \rangle$.

Accepted Answer

$$44/\sqrt{5}$$

Question 7

A particle starts at $$\mathbf{r}(0) = \langle 1, 0, 0 \rangle$$ with initial velocity $$\mathbf{v}(0) = \mathbf{i} - \mathbf{j} + \mathbf{k}$$ and acceleration $$\mathbf{a}(t) = 4t \mathbf{i} + 6t \mathbf{j} + \mathbf{k}. $$What is the position vector $$\mathbf{r}(t)$$?

Accepted Answer

$$\langle 1 + t + 2t^2$, -t + 3t^2$, t + \frac{1}{2}t^2$ \rangle$$

Question 8

Let $$f(x, y) = x^2$ \sin 2y. $$What is the directional derivative of $$f at$$ $$(1, \frac{\pi}{2}) in $$the direction of $$\mathbf{v} = 3\mathbf{i} - 4\mathbf{j}$$?

Accepted Answer

$$-8$$

Question 9

Let $$f(x, y) = 1 - \frac{1}{10}(x^2$ + 4y^2$). $$What is the equation of the tangent plane to the surface $$z = f(x, y) at$$ $$(1, 1)$$?

Accepted Answer

$$z = 0.5 - 0.2(x - 1) - 0.8(y - 1)$$

Question 10

Evaluate the following limit or state why it does not exist: $$\lim\limits_{(x, y) 	o (0, 0)} \left( \frac{x^2$ - y^2$}{x^2$ + y^2$} \$$right)^2$$.

Accepted Answer

The limit does not exist.

Question 11

Let $$f(x, y) = \ln \left( \frac{x}{y} \right). $$Use differentials to estimate the change in $$z as$$ $$(x, y)$$ changes from $$(2, 3) to$$ $$(1.98, 3.01). $$Which of the following is the correct estimate?

Accepted Answer

$$-\frac{1}{100} - \frac{1}{300}$$

Question 12

Find the length of the curve $$\mathbf{r}(t) = \langle \cos t, \sin t, \ln \cos t \rangle$$ for $$0 \leq t \leq \frac{\pi}{4}.$$

Accepted Answer

$$\ln \left| \sec \frac{\pi}{4} + 	an \frac{\pi}{4} ight|$$

Question 13

Let $$f(x, y) = 3xy^2$ - 2y + 5x^2$y^2. $$Find a unit vector in the direction of steepest ascent at $$(1, 1)$.

Accepted Answer

$$\frac{1}{\sqrt{61}} \langle 28, 57 \rangle$$

Question 14

Let $$f(x, y) = 3xy^2$ - 2y + 5x^2$y^2. $$Which of the following is an equation for the tangent plane to $$z = f(x, y) at$$ $$(1, 1)$?

Accepted Answer

$$z = 6 + 28(x - 1) + 57(y - 1)$$

Question 15

Let $$f(x, y) = 3xy^2$ - 2y + 5x^2$y^2. $$Find two different unit vectors that point in directions of no change along the graph of $$f at$$ $$(1, 1)$.

Accepted Answer

$$\frac{1}{\sqrt{61}} \langle 57, -28 \rangle$$ and $$\frac{1}{\sqrt{61}} \langle -57, 28 \rangle$$

Question 16

Let $$f(x, y) = 3xy^2$ - 2y + 5x^2$y^2. $$Verify that $$f_{xy} = f_{yx}$.

Accepted Answer

Both are equal to $$6y + 20xy$$

Question 17

Let $$f(x, y) = 1 - \frac{1}{10}(x^2$ + 4y^2$). $$What is the range of $$f$$?

Accepted Answer

$$(-\infty, 1]$$

Question 18

Find the angle $$	heta$$ between the planes $$x + y - z = 1$$ and $$2x - 3y + 4z = 5. $$Which formula should you use?

Accepted Answer

$$\cos 	heta = \frac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|}$$

Question 19

Let $$f(x, y) = \ln(9 - x^2$ - 9y^2$). At $$which points is the tangent plane to the graph of $$f$$ horizontal?

Accepted Answer

$$(0, 0)$$

Math 210 Midterm I Review – Calculus Study Guidance

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