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Solutions
55 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

`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