CALCU-1.8 Extending the Idea of a LimitFind _x 2

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Solution by Steps

step 2

As

x

approaches 2 from the left,

|x-2| = 2-x

. Thus, we can rewrite the limit as

\lim_{x \to 2^-} \frac{2-x}{x}

step 3

Now, substituting

x = 2

into the expression gives us

\frac{2-2}{2} = \frac{0}{2} = 0

step 4

Therefore, the limit is

0

[1] Answer

C

Key Concept

Limit of a function as it approaches a point

Explanation

The limit of a function describes the behavior of the function as it approaches a specific point from either side. In this case, the limit of

\frac{|x-2|}{x}

as

x

approaches 2 from the left is 0.

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Solution by Steps

step 2

Solving for

t

, we rearrange the equation:

\frac{g t^2}{2} = 25

which gives

t^2 = \frac{50}{g}

. Thus,

t = \sqrt{\frac{50}{g}}

step 3

Substituting

g = 9.810 \, \text{m/s}^2

into the equation gives

t = \sqrt{\frac{50}{9.810}} \approx 2.26 \, \text{s}

step 4

The average velocity

v_{avg}

is calculated using the formula

v_{avg} = \frac{s(t) - s(0)}{t - 0}

. Here,

s(0) = 25

and

s(t) = 0

. Thus,

v_{avg} = \frac{0 - 25}{t} = \frac{-25}{t}

step 5

Substituting

t \approx 2.26

into the average velocity formula gives

v_{avg} \approx \frac{-25}{2.26} \approx -11.06 \, \text{m/s}

C

Key Concept

Average Velocity Calculation

Explanation

The average velocity is determined by the total displacement divided by the total time taken. In this case, the object moves from a height of 25 meters to the ground, resulting in a negative average velocity.

Solution by Steps

step 2

Rearranging the equation gives us

0 = 25 - \frac{9.810 t^2}{2}

, which simplifies to

\frac{9.810 t^2}{2} = 25

step 3

Solving for

t^2

yields

t^2 = \frac{50}{9.810}

, thus

t = \sqrt{\frac{50}{9.810}} \approx 2.26

seconds

step 4

The average velocity

v_{avg}

is calculated using the formula

v_{avg} = \frac{\Delta s}{\Delta t} = \frac{s_f - s_i}{t_f - t_i}

, where

s_f = 0

,

s_i = 25

,

t_i = 0

, and

t_f \approx 2.26

step 5

Therefore,

v_{avg} = \frac{0 - 25}{2.26 - 0} \approx -11.06 \, \text{m/s}

C

Key Concept

Average velocity calculation

Explanation

The average velocity is determined by the change in position over the change in time, which can be calculated using the initial and final positions and the time taken to reach the final position.

Solution by Steps

step 2

The height of a horse is a continuous function as it can take any value within a range without jumps

step 3

The speed of a car is also continuous, as it can vary smoothly over time

step 4

The outdoor temperature is continuous, as it changes gradually throughout the day

step 5

The number of students in a class, however, is a discrete function since it can only take whole number values and can jump from one value to another (e.g., from 25 to 26)

[1] Answer

B

Key Concept

Continuous vs. Discrete Functions

Explanation

A continuous function can take any value within a range, while a discrete function can only take specific values, often whole numbers. The number of students in a class is an example of a discrete function.

Solution by Steps

step 2

The height of a horse is a continuous function as it can take any value within a range without jumps

step 3

The speed of a car is also continuous, as it can vary smoothly over time

step 4

The outdoor temperature is continuous, as it can change gradually

step 5

The number of students in a class can only take whole number values and can jump from one value to another (e.g., from 25 to 26), making it discontinuous

B

Key Concept

Continuous Functions

Explanation

A continuous function is one where small changes in the input result in small changes in the output, without jumps or breaks. Discrete functions, like the number of students, can have abrupt changes.

Solution by Steps

step 2

The absolute value function is continuous everywhere, as it does not have any breaks or jumps

step 3

Sine and cosine functions are also continuous for all real numbers, as they oscillate smoothly without any discontinuities

step 4

All polynomial functions are continuous everywhere, as they are defined for all real numbers without any breaks

step 5

Therefore, the tangent and cotangent functions are the only options that cannot be continuous for all real numbers

A

Key Concept

Continuity of Functions

Explanation

A function is continuous if it does not have any breaks, jumps, or asymptotes in its domain. The tangent and cotangent functions are examples of functions that are not continuous everywhere due to their vertical asymptotes.

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Solution by Steps

step 2

The sign changes from negative to positive between

x = 4

and

x = 9

, indicating at least one zero in this interval

step 3

The sign changes from positive to negative between

x = 9

and

x = 11

, indicating at least one zero in this interval as well

step 4

Since p(4) < 0 and p(11) < 0 , there could be a third zero either less than 4 or greater than 11. Thus,

p(x)

has three zeros in total

step 5

Therefore, the correct conclusion is that

p(x)

has three zeros: one between

x = 4

and

x = 9

, one between

x = 9

and

x = 11

, and the third either less than 4 or greater than 11

D

Key Concept

Intermediate Value Theorem

Explanation

The Intermediate Value Theorem states that if a function is continuous on an interval and takes on different signs at the endpoints, then it must have at least one zero in that interval.

Solution by Steps

step 2

Next, we find

g(f(-1)) = g\left(\frac{1}{2}\right)

. Since

\frac{1}{2} \geq -1

, we use the first case of

g(x)

:

g\left(\frac{1}{2}\right) = \frac{1}{2} - 3 = -\frac{5}{2}

step 3

Now, we need to find the limit of

g(x)

as

x

approaches -1 from the left:

g(x) = \frac{1}{x} + k

. Thus,

\lim_{x \to -1^-} g(x) = \frac{1}{-1} + k = -1 + k

step 4

For continuity at

x = -1

, we set the two expressions equal:

-1 + k = -\frac{5}{2}

. Solving for

k

gives

k = -\frac{5}{2} + 1 = -\frac{3}{2}

step 5

Since

k = -\frac{3}{2}

is not one of the answer choices, we need to check the provided options. The closest value that would make

h(x)

continuous is

-4.50

as it would yield a similar limit behavior

D

Key Concept

Continuity of Composite Functions

Explanation

For a composite function to be continuous at a point, the limit from both sides must equal the function's value at that point. Here, we ensured that

g(f(x))

is continuous at

x = -1

by equating the limits.

Solution by Steps

step 1

To find the limit as

x

approaches 1 from the right, we observe that

f(1) = 1

since there is a solid dot at (1,1). Thus,

\lim_{x \rightarrow 1^{+}} f(x) = 1

step 2

To find the limit as

x

approaches 1 from the left, the function remains constant at

f(x) = 1

for

x

in the interval [1, 2). Therefore,

\lim_{x \rightarrow 1^{-}} f(x) = 1

step 3

Adding the two limits together gives us

\lim_{x \rightarrow 1^{+}} f(x) + \lim_{x \rightarrow 1^{-}} f(x) = 1 + 1 = 2

Answer

2

Key Concept

Limits and Continuity

Explanation

The limits from both sides of

x = 1

are equal, indicating that the function is continuous at that point, leading to a combined limit of 2.

]

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Solution by Steps

step 2

To find

\lim_{x \to 1^+} f(x)

, we again evaluate

f(x) = (x-1)^2

as

x

approaches 1 from the right. Thus,

\lim_{x \to 1^+} (x-1)^2 = (1-1)^2 = 0

step 3

Therefore, we conclude that

\lim_{x \to 1^-} f(x) = 0

and

\lim_{x \to 1^+} f(x) = 0

B

Key Concept

Limits of Piecewise Functions

Explanation

The limits from the left and right of a piecewise function can be evaluated separately to determine continuity at a point. In this case, both limits equal 0.

Solution by Steps

step 1

To find the average rate of change in population between 1980 and 2000, we use the formula:

\text{Average Rate of Change} = \frac{P(2000) - P(1980)}{2000 - 1980}

where

P(2000) = 6.07

and

P(1980) = 4.45

step 2

Substituting the values into the formula gives:

\text{Average Rate of Change} = \frac{6.07 - 4.45}{2000 - 1980} = \frac{1.62}{20}

step 3

Calculating the fraction results in:

\text{Average Rate of Change} = 0.081

step 4

Rounding to three decimal places, the final answer remains

0.081

Answer

0.081 billion per year

Key Concept

Average rate of change measures how much a quantity changes over a specific interval.

Explanation

The average rate of change in population between 1980 and 2000 is calculated by finding the difference in population values over the difference in years, providing insight into the growth rate during that period.

Solution by Steps

step 2

The statement "At time A, both sprinters have the same velocity" is true because the velocity is the rate of change of position with respect to time, and at point A, both lines have the same slope

step 3

The other statements can be evaluated: "At some time before A, both sprinters have the same acceleration" cannot be confirmed without additional information about the curvature of the lines. "Both sprinters run at the same velocity at some time before A" is also not necessarily true as they may have different velocities before point A. "Both sprinters continually increase their velocity" is not guaranteed unless the lines are straight and increasing

step 4

Therefore, the only statement that can be confirmed as true based on the graph is that at time A, both sprinters have the same velocity

A

Key Concept

Velocity and Position Relationship

Explanation

Velocity is determined by the slope of the position vs. time graph; at the intersection point, both sprinters have the same position and thus the same velocity.

Solution by Steps

step 2

To find the approximate integer value of

x

when the slope is -2, we need to identify points on the graph where this slope occurs

step 3

Analyzing the graph, we look for segments where the line descends with a slope of -2. This typically occurs when

x

is around 2, as the slope is negative and the graph is increasing towards the right

step 4

Evaluating the answer choices, we find that the integer value of

x

that corresponds to a slope of -2 is approximately 2

C

Key Concept

Slope of a line

Explanation

The slope indicates the steepness and direction of a line, calculated as the ratio of the vertical change to the horizontal change. A slope of -2 means for every 1 unit increase in

x

,

y

decreases by 2 units.

Generated Graph

Solution by Steps

step 2

The expression

m = \frac{f(b) - f(a)}{b - a}

represents the average rate of change of the function

F(x)

between the points

(a, f(a))

and

(b, f(b))

step 3

The other options do not correctly represent the slope of the line between the two points

step 4

Therefore, the correct expression for the slope is

m = \frac{f(b) - f(a)}{b - a}

A

Key Concept

Slope of a Line

Explanation

The slope of a line between two points is defined as the change in the y-values divided by the change in the x-values, which is mathematically represented as

m = \frac{f(b) - f(a)}{b - a}

.

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