Textbook

Calculus and Vectors McGraw-Hill
Chapter

Chapter 1
Section

Rates of Change, Limits and Continuity Chapter Test

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Solutions
15 Videos

Which of the following functions is defined at `x = 2`

, but is not differentiable at `x =2`

? Give reasons for your choice.

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Q1

Which of the following does not provide the exact value of the instantaneous rate of change at `x = a`

? Explain.

A. ```
\displaystyle
f'(a)
```

B. ```
\displaystyle
\lim_{h \to 0} \frac{f(a+ h) - f(a)}{h}
```

C. ```
\displaystyle
\frac{f(a + h) - f(a)}{h}
```

D. ```
\displaystyle
\lim_{x \to a} \frac{f(x) - f(a)}{x -a}
```

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Q2

Evaluate the limit if it exists.

```
\displaystyle
\lim_{x\to 9}(4x -1)
```

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Q3a

Evaluate the limit if it exists.

```
\displaystyle
\lim_{x \to -3} (2x^4 -3x^2 + 6)
```

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Q3b

Evaluate the limit if it exists.

```
\displaystyle
\lim_{x \to 5} \frac{x^2 -3x - 10}{x - 5}
```

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Q3c

Evaluate the limit if it exists.

```
\displaystyle
\lim_{x \to 0} \frac{9x}{2x^2 -5x}
```

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Q3d

Evaluate the limit if it exists.

```
\displaystyle
\lim_{x \to 7} \frac{x^2 -49}{x - 7}
```

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Q3e

Evaluate the limit if it exists.

```
\displaystyle
\lim_{x \to \infty} \frac{-1}{2 + x^2}
```

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Q3f

Which of the following is not a true statement about limits? Justify your answer.

A. A limit can be used to determine the end behaviour of a graph.

B. A limit can be used to determine the behaviour of a graph on either side of a vertical asymptote.

C. A limit can be used to determine the average rate of change between two points on a graph.

D. .A limit can be used to determine if a graph is discontinuous.

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Q4

a) Use the first principles definition to determine ```
\displaystyle
\frac{dy}{dx}
```

for `y =x^3 -4x^2`

.

b) Sketch `f(x)`

and `f'(x)`

.

c) Determine the equation of the tangent to the function at `x = -1`

.

d) Sketch the tangent on the graph of the function.

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Q5

Determine the following limits for the graph below.

a) ```
\displaystyle
\lim_{ x \to -2 } f(x)
```

b) ```
\displaystyle
\lim_{ x\to 0^-} f(x)
```

c) ```
\displaystyle
\lim_{ x\to 1^-} f(x)
```

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

e) ```
\displaystyle
\lim_{ x\to -4^-} f(x)
```

f) ```
\displaystyle
\lim_{ x\to \infty} f(x)
```

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Q6

Examine the given graph and answer the following questions.

a) State the domain and range of the function.

b) Evaluate each limit for the graph.

i) ```
\displaystyle
\lim_{ x\to +\infty}\frac{3x}{x -4}
```

ii) ```
\displaystyle
\lim_{ x\to -\infty}\frac{3x}{x -4}
```

iii) ```
\displaystyle
\lim_{ x\to 4^+}\frac{3x}{x -4}
```

iv) ```
\displaystyle
\lim_{ x\to 4^-}\frac{3x}{x -4}
```

v) ```
\displaystyle
\lim_{ x\to 6}\frac{3x}{x -4}
```

vi) ```
\displaystyle
\lim_{ x\to -2}\frac{3x}{x -4}
```

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Q7

A carpenter is constructing a large cubical storage shed. The volume of the shed is given by ```
\displaystyle
V(x) = x^2(\frac{16- x^2}{4x})
```

, where x is the side length, in metres.

a) Simplify the expression for the volume of the shed.

b) Determine the average rate of change of the volume of the shed when the side lengths are between 1.5 m and 3 m.

c) Determine the instantaneous rate of change of the volume of the shed when the side length is 3 m.

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Q8

A stone is tossed into a pond, creating a circular ripple on the surface. The radius of the ripple increases at the rate of 0.2 m/s.

a) Determine the length of the radius at the following times.

- i) 1s
- ii) 3s
- iii) 5s

b) Determine an expression for the instantaneous rate of change in the area outlined by the circular ripple with respect to the radius.

c) Determine the instantaneous rate of change of the area corresponding to each radius in part a).

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Q9

Match functions a), b), c), and d) with their corresponding derivative functions A, B, C, and D.

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Q10