Chapter Review Limits and Continuity
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Chapter 1
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Chapter Review Limits and Continuity
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Solutions 43 Videos

Consider the function f(x)=5x^2-8x.

  • Find the slope of the secant that joins on the graph give by x=-2 and x=3.
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2.12mins
Q1a

Consider the function f(x)=5x^2-8x.

  • Determine the average rate of change as x changes from -1 to 4.
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0.59mins
Q1b

Consider the function f(x)=5x^2-8x.

  • Find an equation for the line that is tangent to the graph of the function at x=1.
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0.55mins
Q1c

Calculate the slope of the tangent to the given function at the given point or value of x.

\displaystyle{f(x)=\frac{3}{x+1}}, P(2,1)

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1.45mins
Q2a

Calculate the slope of the tangent to the given function at the given point or value of x.

\displaystyle g(x)=\sqrt{x+2}, P(-1,1)

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1.23mins
Q2b

Calculate the slope of the tangent to the given function at the given point or value of x.

\displaystyle h(x)=\frac{2}{\sqrt{x+5}}, P(4, \frac{2}{3})

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2.52mins
Q2c

Calculate the slope of the tangent to the given function at the given point or value of x.

\displaystyle f(x)=\frac{5}{x-2}, P(4,\frac{5}{2})

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1.12mins
Q2d

Calculate the slope of

f(x)= \begin{cases} 4-x^2, \text{if }\ x\leq 1 \\ 2x+1, \text{if } x> 1 \end{cases}

at each of the following points: P(-1,3)

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1.36mins
Q3a

Calculate the slope of

f(x)= \begin{cases} 4-x^2, \text{if }\ x\leq 1 \\ 2x+1, \text{if } x> 1 \end{cases}

at each of the following points: P(2,5).

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0.19mins
Q3b

The height in metres, of an object that has fallen from a height of 180 m is given by the position function s(t)=-5t^2+180, where t\geq 0 and t is in seconds.

Find the average velocity during each of the first two seconds.

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0.26mins
Q4a

The height in metres, of an object that has fallen from a height of 180 m is given by the position function s(t)=-5t^2+180, where t\geq 0 and t is in seconds.

Find the velocity of the object when t=4.

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1.12mins
Q4b

The height in metres, of an object that has fallen from a height of 180 m is given by the position function s(t)=-5t^2+180, where t\geq 0 and t is in seconds.

At what velocity will the object hit the ground?

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2.00mins
Q4c

After t minutes of growth, a certain bacterial culture has a mass, in grams, of M(t)=t^2.

How much does the bacterial culture grow during the time 3\leq t \leq 3.01?

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0.49mins
Q5a

After t minutes of growth, a certain bacterial culture has a mass, in grams, of M(t)=t^2.

What is its average rate of growth during the time interval 3\leq t \leq 3.01?

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0.24mins
Q5b

After t minutes of growth, a certain bacterial culture has a mass, in grams, of M(t)=t^2.

What is its rate of growth when t=3?

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0.27mins
Q5c

It is estimated that, t years from now, that amount of waste accumulated Q, in tonnes, will be Q(t)=10^4(t^2+15t+70), 0\leq t \leq 10.

How much waste has been accumulated up to now?

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0.23mins
Q6a

It is estimated that, t years from now, that amount of waste accumulated Q, in tonnes, will be Q(t)=10^4(t^2+15t+70), 0\leq t \leq 10.

What will be the average rate of change in this quantity over the next three years?

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1.08mins
Q6b

It is estimated that, t years from now, that amount of waste accumulated Q, in tonnes, will be Q(t)=10^4(t^2+15t+70), 0\leq t \leq 10.

What is the present rate of change in this quantity?

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1.03mins
Q6c

It is estimated that, t years from now, that amount of waste accumulated Q, in tonnes, will be Q(t)=10^4(t^2+15t+70), 0\leq t \leq 10.

When will the rate of change reach 3.0 \times 10^5 per year?

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3.04mins
Q6d

The electrical power p(t), in kilowatts, being used by a household as a function of time t, in hours, in modelled by a graph where t=0 corresponds to 06:00. The graph indicates peak use at 08:00 and a power failure between 09:00 and 10:00.

(a) Determine \lim\limits_{t\to 2}p(t).

(b) Determine \lim\limits_{t\to 4^+}p(t) and \lim\limits_{t\to 4^-}p(t)

(c) For what values of t is p(t) discontinuous?

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0.55mins
Q7

Sketch a graph of any function that satisfies the given conditions.

\lim\limits_{x\to -1}f(x)=0.5, f is discontinuous at x=-1

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0.30mins
Q8a

Sketch a graph of any function that satisfies the given conditions.

f(x)=-4 if x<3, f is an increasing function when x>3, \lim\limits_{x\to 3^+}f(x)=1

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0.47mins
Q8b

Sketch the graph of the following function:

f(x)= \begin{cases} x+1, \text{if } x<-1 \\ -x+1, \text{if } \ -1 \leq x < 1 \\ x-2, \text{if} \ x > 1 \end{cases}

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1.11mins
Q9a

f(x)= \begin{cases} x+1, \text{if } x<-1 \\ -x+1, \text{if } \ -1 \leq x < 1 \\ x-2, \text{if} \ x > 1 \end{cases}

Find all values at which the function is discontinuous.

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0.11mins
Q9b

f(x)= \begin{cases} x+1, \text{if } x<-1 \\ -x+1, \text{if } \ -1 \leq x < 1 \\ x-2, \text{if} \ x > 1 \end{cases}

Find the limits at those values, if they exist.

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0.44mins
Q9c

Determine whether f(x)=\displaystyle{\frac{x^2+2x-8}{x+4}} is continuous at x=-4.

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0.32mins
Q10

Consider the function f(x)=\displaystyle{\frac{2x-2}{x^2+x-2}}

For what values of x is f discontinuous?

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0.26mins
Q11a

Consider the function f(x)=\displaystyle{\frac{2x-2}{x^2+x-2}}

At each point where f is discontinuous, determine the limit of f(x), if it exists.

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0.39mins
Q11b

Evaluate the limit of each difference quotient. Interpret the limit as the slope of the tangent to a curve at a specific point.

\lim\limits_{h\to 0}\displaystyle\frac{(5+h)^2-25}{h}

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0.35mins
Q16a

Evaluate the limit of each difference quotient. Interpret the limit as the slope of the tangent to a curve at a specific point.

\lim\limits_{h\to 0}\displaystyle\frac{\sqrt{4+h}-2}{h}

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0.49mins
Q16b

Evaluate the limit of each difference quotient. Interpret the limit as the slope of the tangent to a curve at a specific point.

\lim\limits_{h\to 0}\displaystyle\frac{\frac{1}{(4+h)}-\frac{1}{4}}{h}

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0.46mins
Q16c

Evaluate the limit, if the limit exists.

\lim\limits_{x\to -4} \displaystyle\frac{x^2+12x+32}{x+4}

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0.29mins
Q17a

Evaluate the limit, if the limit exists.

\lim\limits_{x\to a} \displaystyle\frac{(x+4a)^2-25a^2}{x-a}

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0.46mins
Q17b

Evaluate the limit, if the limit exists.

\lim\limits_{x\to 0} \displaystyle\frac{\sqrt{x+5}-\sqrt{5-x}}{x}

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1.05mins
Q17c

Evaluate each limit using one of the algebraic methods discussed in this chapter, if the limit exists.

\lim\limits_{x\to 2} \displaystyle\frac{x^2-4}{x^3-8}

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0.30mins
Q17d

Evaluate the limit, if the limit exists.

\lim\limits_{x\to 4} \displaystyle\frac{4-\sqrt{12+x}}{x-4}

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1.08mins
Q17e

Evaluate the limit, if the limit exists.

\lim\limits_{x\to 0} \displaystyle\frac{1}{x}\left(\frac{1}{2+x}-\frac{1}{2}\right)

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0.29mins
Q17f

Explain why the given limit does not exist.

\lim\limits_{x\to 3} \sqrt{x-3}

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0.42mins
Q18a

Explain why the given limit does not exist.

\displaystyle \lim_{x\to 2} \displaystyle\frac{x^2-4}{x^2-4x+4}

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1.00mins
Q18b

Explain why the \lim\limits_{x\to 1} f(x) does not exist.

f(x)=\begin{cases} -5, &\text{if }x < 1 \\ 2, &\text{if } x\geq 1 \end{cases}

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0.32mins
Q18c

Explain why the given limit does not exist.

\lim\limits_{x\to 2} \displaystyle\frac{1}{\sqrt{x-2}}

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0.24mins
Q18d

Explain why the given limit does not exist.

\lim\limits_{x\to 0} \displaystyle\frac{|x|}{x}

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0.38mins
Q18e

Explain why the \lim\limits_{x\to -1} f(x) does not exist.

f(x)= \begin{cases} 5x^2, &\text{if } x < -1 \\ 2x+1, &\text{if } x \geq -1 \end{cases}

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0.34mins
Q18f