Textbook

Calculus and Vectors McGraw-Hill
Chapter

Chapter 3
Section

Chapter Review Graphing and Optimization

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Solutions
26 Videos

Find the increasing and decreasing intervals for the function.

```
\displaystyle
f(x) = 7 + 6x-x^2
```

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Q1a

Find the increasing and decreasing intervals for the function.

```
\displaystyle
f(x) = x^3 -48x + 5
```

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Q1b

Find the increasing and decreasing intervals for the function.

```
\displaystyle
f(x) = x^4 -18x^2
```

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Q1c

Find the increasing and decreasing intervals for the function.

```
\displaystyle
f(x) = x^3 + 10x -9
```

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Q1d

The first derivative of a function is `f(x) = x(x - 3)^2`

; Make a table showing the
increasing and decreasing intervals of the function.

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Q2

Find the local extrema for each function and classify them as local maxima or local minima.

```
\displaystyle
y = 3x^2 +24x -8
```

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Q3a

Find the local extrema for each function and classify them as local maxima or local minima.

```
\displaystyle
y = 16-x^4
```

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Q3b

Find the local extrema for each function and classify them as local maxima or local minima.

```
\displaystyle
y = x^3 + 9x^2 -21x -12
```

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Q3c

The speed, in kilometres per hour, of a certain car t seconds after passing a police radar location is given by the function

```
\displaystyle
v(t) = 3t^2 -24t + 88
```

a) Find the minimum speed of the car.

b) The radar tracks the car on the interval `2 < t < 5`

. Find the maximum speed of the car on this interval.

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Q4

Find and classify all critical points of the function `f(x) =x^3-8x2+5x +2`

on the interval `0\leq x \leq 6`

.

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Q5

Which of these statements is most accurate regarding the number of points of inflection on a cubic function? Explain.

A. There is at least one point of inflection.

B. There is exactly one point of inflection.

C. There are no points of inflection.

D. It is impossible to tell.

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Q6

A polynomial function of degree four (3 quartic function) has either no points of inflection or two points of inflection. Is this statement true or false? Explain your reasoning.

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Q7

For the function `f(x)=x^4-2x^3 -12x^2+3`

, determine the points of inflection and the
intervals of concavity.

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Q8

The graph shows the first derivatlve, `f(x)`

, of a function `f(x)`

. Copy the graph into your notebook.

a) Sketch a possible graph of `f(x)`

.

b) Sketch a possible graph of the second derivative `f''(x)`

.

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Q9

For the function `f(x) = 2x^3 - x^4`

, determine the critical points and classify them using the second derivative test. Sketch the function.

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Q10

For each function, state equations for any vertical asymptotes.

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

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Q11a

For each function, state equations for any vertical asymptotes.

```
\displaystyle
f(x) = \frac{2x -3}{2x -4}
```

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Q11b

For each function, state equations for any vertical asymptotes.

```
\displaystyle
f(x) = \frac{x^2 + 1}{x^2 -3x - 10}
```

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Q11c

For each function, state equations for any vertical asymptotes.

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

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Q11d

Consider the function ```
\displaystyle
f(x) = \frac{x + 4}{x^2}
```

.

a) Without graphing, evaluate ```
\displaystyle
\lim_{ x\to 0^+} f(x)
```

.

b) How can you use your answer to part a) to evaluate `\lim_{x \to 0^+}f(x)`

? Explain your reasoning.

c) State the coordinates of the x-intercepts.

d) Find the coordinates of the turning point using the first derivative.

e) Use a graphing calculator to verify your results.

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Q12

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

.

a) Determine whether 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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Q13

Analyse and sketch the function.

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

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Q14a

Analyse and sketch the function.

```
\displaystyle
f(x) = \frac{1}{4}x^4 - \frac{9}{2}x^2
```

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Q14b

Analyse and sketch the function.

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

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Q14c

The concentration, in milligrams per cubic centimetre, of a particular drug in a patient's bloodstream is given by the formula ```
\displaystyle
C(t) = \frac{0.12t}{t^2 +2t + 2}
```

, where `t`

represents th number of hours after the drug is taken.

a) Find the maximum concentration on the interval `0 \leq t \leq 4`

.

b) Determine when the maximum concentration occurs.

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Q15

An openâ€”top box is to be constructed so that its base is twice as long as it is wide. Its volume is to be 2400 `cm^3`

.
Find the dimensions that will minimize the amount of cardboard required.

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Q16