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Solutions
38 Videos

For each function, determine the coordinates of the local extrema. Classify each point as a local maximum or a local minimum.

`f(x)=x^3 - 6x`

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1.29mins

Q1a

For each function, determine the coordinates of the local extrema. Classify each point as a local maximum or a local minimum.

`g(x)=-4x^4+2x^2`

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4.04mins

Q1b

For each function, determine the coordinates of the local extrema. Classify each point as a local maximum or a local minimum.

`f(x)=-x^3+3x-2`

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3.01mins

Q1c

`h(x)= 2x^2 + 4x + 5`

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0.36mins

Q1d

For each function, determine the coordinates of any points of inflection.

```
\displaystyle
f(x)= 2x^3-4x^2
```

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1.21mins

Q2a

For each function, determine the coordinates of any points of inflection.

```
\displaystyle
f(x)= x^5 + 20x^2 + 5
```

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1.01mins

Q2b

For each function, determine the coordinates of any points of inflection.

`f(x)=x^5-30x^3`

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4.14mins

Q2c

For each function, determine the coordinates of any points of inflection.

`f(x)=3x^5-5x^4-40x^3+120x^2`

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1.12mins

Q2d

Consider the function `f(x)=x^4-8x^3`

.

a) Determine if the function is even, odd, or neither.

b) Determine the domain of the function.

c) Determine the intercepts

d) Find and classify the critical points. Identify the intervals of increase and decrease, and state the intervals of concavity.

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3.23mins

Q3

Sketch each function.

`f(x)=x^3+1`

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0.42mins

Q4a

Sketch each function.

`h(x)=x^5+20x^2+5`

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12.18mins

Q4b

Sketch each function.

`k(x)=\displaystyle{\frac{1}{2}}x^4-2x^3`

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3.17mins

Q4c

Sketch each function.

`b(x)=-(2x-1)(x^2-x-2)`

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3.25mins

Q4d

Consider the function
`f(x)=2x^3-3x^2-72x+7`

- What is the maximum number of local extrema this function can have? Explain.
- What is the maximum number of points of inflection this function can have? Explain.
- Find and classify the critical points. Identify the intervals of increase and decrease, and state the intervals of concavity.
- Sketch the function.

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6.34mins

Q5

Do a full sketch by showing how you got

- Horizontal and vertical Asymptotes
- x-intercepts
- Any critical points (max/min)
- Intervals of increase and decrease
- Points of inflection and concavity

```
\displaystyle
h(x) = 3x^2 -27
```

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Q6i

Do a full sketch by showing how you got

- Horizontal and vertical Asymptotes
- x-intercepts
- Any critical points (max/min)
- Intervals of increase and decrease
- Points of inflection and concavity

```
\displaystyle
f(x) = x^5 -2x^4 + 3
```

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Q6ii

Do a full sketch by showing how you got

- Horizontal and vertical Asymptotes
- x-intercepts
- Any critical points (max/min)
- Intervals of increase and decrease
- Points of inflection and concavity

`f(x)= 3x^3 +7x^2 + 3x -1`

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Q9a

Do a full sketch by showing how you got

- Horizontal and vertical Asymptotes
- x-intercepts
- Any critical points (max/min)
- Intervals of increase and decrease
- Points of inflection and concavity

`f(x)= 2x^3 -12x^2 + 18x -4`

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Q9b

Do a full sketch by showing how you got

- Horizontal and vertical Asymptotes
- x-intercepts
- Any critical points (max/min)
- Intervals of increase and decrease
- Points of inflection and concavity

`f(x)= 2x^4 -26x^2 +72`

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Q9c

Do a full sketch by showing how you got

- Horizontal and vertical Asymptotes
- x-intercepts
- Any critical points (max/min)
- Intervals of increase and decrease
- Points of inflection and concavity

`h(x)= 5x^3 -3x^5`

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Q9d

Write an equation for a function that has a local minimum when `x = 2`

and a point of inflection at `x = 0`

.

Explain how these conditions helped you determine the function.

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Q12

Joshua was sketching the graph of the functions f(x) and g(x). He lost part of his notes, including the equations for the functions. Given the partial information below, sketch each function.

The domain of `f`

is `\mathbb{R}`

.

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

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

The y-intercept is 2. The x-intercepts are -1, 3, and 5. There is a local extrema at x = 3.

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Q13a

Joshua was sketching the graph of the functions f(x) and g(x). He lost part of his notes, including the equations for the functions. Given the partial information below, sketch each function.

The polynomial function g has domain `\mathbb{R}`

.

```
\displaystyle
\lim_{x\to -\infty} g(x) = \infty
```

`g`

is an odd function.

There are five x-intercepts, two of which are 4 and -22. The function passes through
`( 1, 5)`

, which is close to a local extremum.

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Q13b

For each function,

i) How does the function behave as `x\to \infty`

? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

```
\displaystyle
g(x) = x^3 -27x
```

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Q18a

For each function,

i) How does the function behave as `x\to \infty`

? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

```
\displaystyle
y = x^4 - 8x^2 + 16
```

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Q18b

For each function,

i) How does the function behave as `x\to \infty`

? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iii) Use the derivative to find the critical point(s). Classify them using the second derivative test.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

```
\displaystyle
k(x) = \frac{1}{1 -x^2}
```

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Q18c

For each function,

i) How does the function behave as `x\to \infty`

? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

```
\displaystyle
f(x) = \frac{x}{x^2 + 1}
```

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Q18d

For each function,

i) How does the function behave as `x\to \infty`

? Explain your reasoning.

ii) Does the function have any symmetry? Explain.

iv) Find the point(s) of inflection and make a table showing the intervals of concavity.

`h(x) = \frac{x-4}{x^2}`

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Q18e

Lectures
2 Videos

**Full Graphing ex1**

Sketch `f(x) = x^4 -8x^3`

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10.22mins

Full Graphing ex1

**Full Graphing ex2**

Sketch ```
\displaystyle
f(x) =x^{\frac{2}{3}}(6-x)^{\frac{1}{3}}
```

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13.00mins

Full Graphing ex2