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For each of the following graphs, state

**i)** the intervals where the function is increasing

**ii)** the intervals where the function is decreasing

**iii)** the points where the tangent to the function is horizontal

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0.25mins

Q1a

For each of the following graphs, state

**i)** the intervals where the function is increasing

**ii)** the intervals where the function is decreasing

**iii)** the points where the tangent to the function is horizontal

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1.10mins

Q1b

Is it always true that an increasing function is concave up in shape? Explain.

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0.13mins

Q2

Determine the critical points for each function. Determine whether the critical point is a local maximum or local minimum and whether or not the tangent is parallel to the `x`

-axis.

`f(x)=-2x^3+9x^2+20`

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1.02mins

Q3a

Determine the critical points for each function. Determine whether the critical point is a local maximum or local minimum and whether or not the tangent is parallel to the `x`

-axis.

`f(x)=x^4-8x^3+18x^2+6`

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1.10mins

Q3b

Determine the critical points for each function. Determine whether the critical point is a local maximum or local minimum and whether or not the tangent is parallel to the `x`

-axis.

`h(x)=\displaystyle{\frac{x-3}{x^2+7}}`

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1.35mins

Q3c

`x`

-axis.

`g(x)=(x-1)^{\displaystyle{\frac{1}{3}}}`

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0.55mins

Q3d

The graph of the function `y=f(x)`

has local extrema at points `A`

, `C`

, and `E`

and points of inflection at `B`

and `D`

. If `a`

,`b`

,`c`

,`d`

, and `e`

are the `x`

-coordinates of the points, state the intervals on which the following conditions are true:

`f'(x)>0`

and `f''(x)>0`

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0.29mins

Q4a

The graph of the function `y=f(x)`

has local extrema at points `A`

, `C`

, and `E`

and points of inflection at `B`

and `D`

. If `a`

,`b`

,`c`

,`d`

, and `e`

are the `x`

-coordinates of the points, state the intervals on which the following conditions are true:

`f'(x)>0`

and `f''(x)<0`

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0.17mins

Q4b

The graph of the function `y=f(x)`

has local extrema at points `A`

, `C`

, and `E`

and points of inflection at `B`

and `D`

. If `a`

,`b`

,`c`

,`d`

, and `e`

are the `x`

-coordinates of the points, state the intervals on which the following conditions are true:

`f'(x)<0`

and `f''(x)>0`

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0.25mins

Q4c

`y=f(x)`

has local extrema at points `A`

, `C`

, and `E`

and points of inflection at `B`

and `D`

. If `a`

,`b`

,`c`

,`d`

, and `e`

are the `x`

-coordinates of the points, state the intervals on which the following conditions are true:

`f'(x)<0`

and `f''(x)<0`

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0.18mins

Q4d

For the following, check for discontinuities and state the equation of any vertical asymptotes. Conduct a limit test to determine the behaviour of the curve on either side of the asymptote.

`y=\displaystyle{\frac{2x}{x-3}}`

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1.26mins

Q5a

For the following, check for discontinuities and state the equation of any vertical asymptotes. Conduct a limit test to determine the behaviour of the curve on either side of the asymptote.

`g(x)=\displaystyle{\frac{x-5}{x+5}}`

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1.16mins

Q5b

Determine the point of inflection on the curve defined by `y=x^3+5`

. Show that the tangent line at this point crosses the curve. Explain.

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0.40mins

Q6

Sketch a graph of a function that is differentiable on the interval `-3\leq x \leq 5`

and satisfies the following conditions:

**i)** There are local maxima at `(-2,10)`

and `(3,4)`

.

**ii)** The function `f`

is decreasing on the intervals `-2 < x < 1`

and `3\leq x \leq 5`

.

**iii)** The derivative `f'(x)`

is positive for `-3 \leq x < -2`

and for `1 < x < 3`

.

**iv)** `f(1)=-6`

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2.12mins

Q7

Each of the following graphs represents the second derivative, `g''(x)`

, of a function `g(x)`

:

**i)** On what interval is the graph of `g(x)`

concave up? on what interval is the graph concave down?

**ii)** List the `x`

-coordinates of the points of inflection.

**iii)** Make a rough sketch of a possible graph for `g(x)`

, assuming that `g(0)=-3`

.

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1.45mins

Q8a

Each of the following graphs represents the second derivative, `g''(x)`

, of a function `g(x)`

:

**i)** On what interval is the graph of `g(x)`

concave up? on what interval is the graph concave down?

**ii)** List the `x`

-coordinates of the points of inflection.

**iii)** Make a rough sketch of a possible graph for `g(x)`

, assuming that `g(0)=-3`

.

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1.43mins

Q8b

If the graph of the function `g(x)=\displaystyle{\frac{ax+b}{(x-1)(x-4)}}`

has a horizontal tangent at point `(2,-1)`

, determine the values of `a`

and `b`

.

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1.59mins

Q9a

a) If the graph of the function `g(x)=\displaystyle{\frac{ax+b}{(x-1)(x-4)}}`

has a horizontal tangent at point `(2,-1)`

, determine the values of `a`

and `b`

.

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1.22mins

Q9b

Sketch each function using suitable techniques.

`y=x^4-8x^2+7`

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4.37mins

Q10a

Sketch each function using suitable techniques.

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

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1.27mins

Q10b

Sketch each function using suitable techniques.

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

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1.32mins

Q10c

Sketch each function using suitable techniques.

`y=x(x-4)^3`

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1.14mins

Q10d

Sketch each function using suitable techniques.

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

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1.32mins

Q10e

Sketch each function using suitable techniques.

`f(t)=\displaystyle{\frac{t^2-3t+2}{t-3}}`

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2.39mins

Q10f

Determine the conditions on parameter `k`

such that the function `f(x)=\displaystyle{\frac{2x+4}{x^2-k^2}}`

will have critical points.

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1.46mins

Q11a

When `k=0, 2`

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

sketch the selection of the curve that lies in the domain `|x|\leq k`

.

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1.16mins

Q11b

Determine the equation of the oblique asymptote in the form `y=mx+b`

for each function, and then show that `\lim\limits_{x\to + \infty}[y-f(x)]=0`

`f(x)=\displaystyle{\frac{2x^2-7x+5}{2x-1}}`

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1.09mins

Q12a

Determine the equation of the oblique asymptote in the form `y=mx+b`

for each function, and then show that `\lim\limits_{x\to + \infty}[y-f(x)]=0`

`f(x)=\displaystyle{\frac{4x^3-x^2-15x-50}{x^2-3x}}`

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1.40mins

Q12b

Determine the critical numbers and the intervals on which `g(x)=(x^2-4)^2`

is increasing or decreasing.

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1.10mins

Q13

Use the second derivative test to identify all maximum and minimum values of `f(x)=x^3+\displaystyle{\frac{3}{2}}x^2-7x+5`

on the interval `-4\leq x \leq 3`

.

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1.07mins

Q14

Use the `y`

-intercept, local extrema, intervals of concavity, and points of inflection to graph `f(x)=4x^3+6x^2-24x-2`

.

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2.48mins

Q15

Let `p(x)=\displaystyle{\frac{3x^3-5}{4x^2+1}}`

, `q(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}`

, `r(x)=\displaystyle{\frac{x^2-2x-8}{x^2-1}}`

, and `s(x)=\displaystyle{\frac{x^3+2x}{x-2}}`

Determine the asymptotes for each function, and identify their type (vertical, horizontal, or oblique).

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1.06mins

Q16a

Let `p(x)=\displaystyle{\frac{3x^3-5}{4x^2+1}}`

, `q(x)=\displaystyle{\frac{3x-1}{x^2-2x-3}}`

, `r(x)=\displaystyle{\frac{x^2-2x-8}{x^2-1}}`

, and `s(x)=\displaystyle{\frac{x^3+2x}{x-2}}`

Graph `y=r(x)`

, showing clearly the asymptotes and the intercepts.

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2.25mins

Q16b

If `f(x)=\displaystyle{\frac{x^3+8}{x}}`

, determine the domain, intercepts, asymptotes, intervals of increase and decrease, and concavity. Locate any critical points and points of inflection. Use this information to sketch the graph of `f(x)`

. Which of the following is the graph of `f(x)`

.

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6.12mins

Q17

Given the graph of `y=f''(x)`

, Which of the following is a possible graph of the original function, `y=f(x)`

.

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0.53mins

Q18

For `f(x)=\displaystyle{\frac{5x}{(x-1)^2}}`

, show that `f'(x)=\displaystyle{\frac{-5(x+1)}{(x-1)^3}}`

and `f''(x)=\displaystyle{\frac{100(x+2)}{(x-1)^4}}`

. Use the function and its derivatives to determine the domain, intercepts, asymptotes, intervals of increase and decrease, and concavity, and to locate any local extrema and points of inflection. Use this information to sketch the graph of `f`

.

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5.55mins

Q19

The graphs of a function and its derivatives, `y=f(x)`

, `y=f'(x)`

, and `y=f''(x)`

, are shown on each pair of axes. Which is which? Explain how you can tell.

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0.34mins

Q20a

`y=f(x)`

, `y=f'(x)`

, and `y=f''(x)`

, are shown on each pair of axes. Which is which? Explain how you can tell.

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1.02mins

Q20b