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Questions and Answers
What is the average velocity of the car on the interval [1, 1.5]?
What is the limit of the velocity function as h approaches 0 for the expression (s(1 + h) - s(1))/h?
Which of the following limits represents the behavior of the function 2x + 1 as x approaches 1?
What is the value of limx→0 |x| / x?
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In the context of one-sided limits, what does limx→a+ f(x) = L imply?
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What does the notation $\lim_{x \to a} f(x) = L$ signify?
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Which of the following is true about one-sided limits?
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When can L'Hôpital's Rule be applied?
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Which statement best describes continuity at a point $a$?
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What is the result of applying the Sum Law of limits?
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Study Notes
Velocity and Position
- Average Velocity: The change in position over change in time.
- (s(b) - s(a)) / (b - a)
- Using the position function s(t) = t2, we can calculate the average velocity over shorter time intervals to approximate the instantaneous velocity at a specific time.
- This involves shortening the time interval.
Limits
- A limit is a value the function approaches as the input approaches a certain value.
- This is denoted as limx→a f(x) = L, which means as x approaches a, f(x) gets very close to L.
One-Sided Limits
- A one-sided limit specifies the direction from which the input approaches the limit point.
- limx→a+ f(x) = L means f(x) approaches L from the right side of a.
- limx→a- f(x) = L means f(x) approaches L from the left side of a.
- If the limit exists, it means the left-side limit and the right-side limit are equal:
- limx→a+ f(x) = limx→a- f(x) = L
- Therefore, limx→a f(x) = L
Examples
- limh→0 (s(1 + h) - s(1))/h = 2
- limx→1 2x+1 = 3
- limx→-1 2x+1 = 3
- limx→1- 2x+1 = 3
- limx→1 x2= 1
- limx→-1 x2 = 1
- limx→1 x2= 1
- limx→2 (x-2)/(x2-4) = 1/4
- limx→2 (x-2)/(x2-4) = 1/4
- limx→2 (x-2)/(x2-4) = 1/4
- limx→∞ 1/x2 = +∞
- limx→0 1/x2 = +∞
- limx→0- 1/x2 = +∞
- limx→0+ 1/x2 = +∞
- limx→0 |x| / x = DNE (Does Not Exist)
- limx→0+ |x| / x = 1
- limx→0- |x| / x = -1
Limits
- Limits describe how a function behaves as its input approaches a specific value.
- The notation ( \lim_{x \to a} f(x) = L ) means ( f(x) ) gets closer to ( L ) as ( x ) gets closer to ( a ).
- Limits can be one-sided, considering values of ( x ) approaching ( a ) from the right ( ( \lim_{x \to a^+} f(x) ) ) or the left ( ( \lim_{x \to a^-} f(x) ) ).
- Limits can be infinite, represented by ( \lim_{x \to a} f(x) = \infty ) or ( -\infty ), indicating a vertical asymptote.
- Several limit laws help simplify calculations:
- Sum Law adds limits of individual functions.
- Product Law multiplies limits of individual functions.
- Quotient Law divides limits of individual functions (provided the denominator's limit is not zero).
- Finding limits typically involves substitution, factoring, rationalization, or L'Hôpital's Rule.
- L'Hôpital's Rule applies when a limit results in an indeterminate form like 0/0 or ∞/∞, allowing the evaluation of the limit of the ratio of derivatives.
- Common limits include:
- ( \lim_{x \to 0} \frac{\sin x}{x} = 1 ).
- ( \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} ).
- ( \lim_{x \to \infty} \frac{1}{x} = 0 ).
Continuity
- A function is continuous at a point ( a ) if three conditions are met:
- The function value ( f(a) ) is defined.
- The limit of the function as ( x ) approaches ( a ) exists.
- The limit equals the function value: ( \lim_{x \to a} f(x) = f(a) ).
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Description
Explore the concepts of average velocity and limits in this quiz. Learn how to calculate average velocities using position functions and understand one-sided limits. Test your knowledge and comprehension of these fundamental calculus ideas.
