Q: Which of the following statements about continuity is TRUE?

A function is continuous at $$x=a if$$ $$\lim_{x \to a} f(x) = f(a).$$

Q: Which of the following is the correct dot product of $$\vec{a} = (1,2,3)$$ and $$\vec{b} = (4,0,-1)$$?

$$1 \times 4 + 2 \times 0 + 3 \times (-1) = 4 - 3 = 1$$

Question 1

A company finds that if it sets the price of a product at $$10, it will sell 100 units per day. For every 1 $$increase in price, it will sell 2 fewer units per day. What price should the company set to maximize its daily revenue?

Accepted Answer

$$\35$

Question 2

Evaluate the following limit: $$\displaystyle \lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 3

Which of the following statements about continuity is TRUE?

Accepted Answer

A function is continuous at $$x=a if$$ $$\lim_{x 	o a} f(x) = f(a).$$

Question 4

Let $$f(x) = x^3$ - 3x^2$ + 2. $$Find all critical points of $$f(x).$$

Accepted Answer

$$x = 1$$ and $$x = 2$$

Question 5

Given $$f(x) = \sqrt{x^2$ + x}(x^2$ + x)$$, find $$f'(x)$$ using the product rule.

Accepted Answer

$$(2x+1)\sqrt{x^2$ + x} + (x^2$ + x)\frac{2x+1}{2\sqrt{x^2$ + x}}$$

Question 6

A ball is thrown vertically upward with position function $$s(t) = -16t^2$ + 64t (in $$feet, $$t in $$seconds). What is the maximum height reached by the ball?

Accepted Answer

$$128$$ feet

Question 7

Which of the following is the correct derivative of $$f(x) = \sin(x)\cos(x)$$?

Accepted Answer

$$\cos^2$(x) - \sin^2$(x)$$

Question 8

Let $$f(x) = \frac{x^2$ + 1}{x}. $$Using the quotient rule, what is $$f'(x)$$?

Accepted Answer

$$\frac{x^2$ - 1}{x^2$}$$

Question 9

Which of the following best describes a removable discontinuity?

Accepted Answer

The function is undefined at a point, but the limit exists and is finite.

Question 10

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 11

A vector in $$\mathbb{R}^3 is $$given by $$\vec{a} = (2, -1, 3). $$What is the magnitude of $$\vec{a}$$?

Accepted Answer

$$\sqrt{2^2$ + (-1)^2$ + 3^2$} = \sqrt{14}$$

Question 12

Let $$f(x) = \ln(x^2$ + 1). $$What is $$f'(x)$$?

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 13

Which of the following is the correct equation of the plane passing through the point $$(1,2,3)$$ with normal vector $$\vec{n} = (2, -1, 4)$$?

Accepted Answer

$$2(x-1) - (y-2) + 4(z-3) = 0$$

Question 14

Given $$f(x) = (x^2$ - x - 1$$)^{100}$$, use the chain rule to find f$$'(x)$$.

Accepted Answer

$$100(x^2 - x$$ - $$1)^{99}(2x - 1$$)$$

Question 15

Which of the following is the correct dot product of $$\vec{a} = (1,2,3)$$ and $$\vec{b} = (4,0,-1)$$?

Accepted Answer

$$1 	imes 4 + 2 	imes 0 + 3 	imes (-1) = 4 - 3 = 1$$

Question 16

Which of the following is the correct cross product $$\vec{a} 	imes \vec{b}$$ for $$\vec{a} = (1,0,0)$$ and $$\vec{b} = (0,1,0)$$?

Accepted Answer

$$(0,0,1)$$

Question 17

Which of the following best describes the concavity of a function $$f(x) at a $$point where $$f''(x) \u003e 0$$?

Accepted Answer

The function is concave up at that point.

Question 18

A line in $$\mathbb{R}^3$$ passes through the point $$(1,2,3)$$ and has direction vector $$(4,5,6). $$Which of the following is its vector equation?

Accepted Answer

$$\vec{r} = (1,2,3) + t(4,5,6)$$

Question 19

Let $$f(x) = |x|. $$Which of the following statements is TRUE?

Accepted Answer

$$f(x) is $$continuous everywhere but not differentiable at $$x=0.$$

MAT1339 Calculus and Vectors: Structured Study Notes

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