Q: Which of the following is the correct definition of continuity at a point $$a$$?

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

Q: Which of the following is the correct definition of the limit from the right at $$a$$?

$$\lim_{x \to a^+} f(x) is $$the value $$f(x)$$ approaches as $$x$$ approaches $$a$$ from values greater than $$a.$$

Question 1

Which of the following best describes the domain of the function $$f(x) = \sqrt{x-2}$$?

Accepted Answer

$$x \geq 2$$

Question 2

Given the function $$f(x) = \frac{1}{x-3}$$, which of the following statements is true about its range?

Accepted Answer

The range is $$(-\infty, 0) \cup (0, \infty)$$

Question 3

If $$f(x) = x^2$ and g(x$$) = \sin(x)$$, what is (f$$ \circ g)(\frac{\pi}{2})$$?

Accepted Answer

$$1$$

Question 4

Which of the following is the correct definition of the vertical line test for functions?

Accepted Answer

A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Question 5

Which of the following is the inverse of the exponential function $$f(x) = e^x$$?

Accepted Answer

$$f^{-1}(x$$) = \ln(x)$$

Question 6

A patient’s temperature $$T (in$$ °C) is modeled by $$T(t) = 37 + 2\sin(\frac{\pi t}{12})$$, where $$t is $$hours after midnight. What is the patient’s temperature at 6 AM?

Accepted Answer

$$T(6) = 37$$

Question 7

Which of the following is NOT a property of logarithmic functions?

Accepted Answer

$$\log(a+b) = \log(a) + \log(b)$$

Question 8

Which of the following is the correct definition of a limit at a point $$a$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that if $$0 \u003c |x-a| \u003c \delta$$, then $$|f(x) - L| \u003c \epsilon.$$

Question 9

A chemical reaction rate $$R(t) is $$modeled by $$R(t) = 5e$$^{-0.2t}$$, where t$ is time in minutes. What is the initial rate?

Accepted Answer

$$5$$

Question 10

Which of the following is true about the function $$f(x) = \frac{x^2$-4}{x-2}$$?

Accepted Answer

It is undefined at $$x=2.$$

Question 11

Which of the following is the correct definition of continuity at a point $$a$$?

Accepted Answer

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

Question 12

A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = 3x^2$ - 12x + 20$$, where $$x is $$the number of units sold. What is the minimum profit?

Accepted Answer

$$P(2) = 8$$

Question 13

Which of the following is the correct definition of the horizontal line test?

Accepted Answer

A function is one-to-one if and only if every horizontal line intersects its graph at most once.

Question 14

Which of the following is the correct formula for the sum of two functions $$f$$ and $$g$$?

Accepted Answer

$$(f+g)(x) = f(x) + g(x)$$

Question 15

If $$f(x) = |x|$$, which of the following describes its graph?

Accepted Answer

It is V-shaped and symmetric about the y-axis.

Question 16

Which of the following is the correct definition of the natural logarithm function?

Accepted Answer

$$\ln(x) is $$the inverse of $$e^x.$$

Question 17

A medication’s concentration $$C(t) in $$the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is hours after administration. What is the concentration after 2 hours?

Accepted Answer

$$C(2) = 10e$$^{-1}$$

Question 18

Which of the following is the correct definition of a one-to-one function?

Accepted Answer

A function $$f is $$one-to-one if $$f(x_1$) = f(x_2$)$$ implies $$x_1$$ = $$x_2.$$

Question 19

Which of the following is the correct definition of the limit from the right at $$a$$?

Accepted Answer

$$\lim_{x 	o a^+} f(x) is $$the value $$f(x)$$ approaches as $$x$$ approaches $$a$$ from values greater than $$a.$$

Calculus I: Course Syllabus and Key Concepts Overview

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