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Solutions
25 Videos

For the function, state whether the value of the second derivative is positive or negative at each of points A, B, C, and D.

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0.22mins

Q1a

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0.37mins

Q1b

Determine the critical points for each function, and use the second derivative test to decide if the point is a local maximum, a local minimum, or neither.

`y =x^3-6x^2-15x + 10`

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0.51mins

Q2a

Determine the critical points for each function, and use the second derivative test to decide if the point is a local maximum, a local minimum, or neither.

`y = \displaystyle{\frac{25}{x^2 + 48}}`

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2.25mins

Q2b

Determine the critical points for each function, and use the second derivative test to decide if the point is a local maximum, a local minimum, or neither.

`s =t+t^{-1}`

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0.58mins

Q2c

`y = (x - 3)^3 + 8`

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0.33mins

Q2d

Determine the points of inflection for each function. Then conduct a test to determine the change of sign in the second derivative.

`y =x^3-6x^2-15x + 10`

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0.59mins

Q3a

Determine the points of inflection for each function. Then conduct a test to determine the change of sign in the second derivative.

`y = \displaystyle{\frac{25}{x^2 + 48}}`

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3.44mins

Q3b

Determine the points of inflection for each function. Then conduct a test to determine the change of sign in the second derivative.

`s =t+t^{-1}`

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1.30mins

Q3c

`y = (x - 3)^3 + 8`

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0.58mins

Q3d

Determine the value of the second derivative at the value indicated. State whether the curve lies above or below the tangent at this point.

`f(x) =2x^3-10x + 3, \text{ at x =2}`

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0.33mins

Q4a

Determine the value of the second derivative at the value indicated. State whether the curve lies above or below the tangent at this point.

` y = x^2 -\displaystyle{\frac{1}{x}}, \text{ at x = -1}`

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1.43mins

Q4b

Determine the value of the second derivative at the value indicated. State whether the curve lies above or below the tangent at this point.

`p(w) = \displaystyle{\frac{w}{\sqrt{w^2 + 1}}}, \text{ at w = 3}`

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2.00mins

Q4c

`s(t) = \displaystyle{\frac{2t}{t -4}}, \text{ at t = -2}`

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1.07mins

Q4d

The graphs represents the second derivative, `f''(x)`

, of a function `f(x)`

:

For each of the graphs above, answer the following questions:

**i)** On which intervals is the graph of `f(x)`

concave up? On which intervals is the graph concave down?

**ii)** List the x-coordinates of all the points of inflection.

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

, assuming that `f(0) = 2`

.

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1.06mins

Q5a

The graphs represents the second derivative, `f''(x)`

, of a function `f(x)`

:

For each of the graphs above, answer the following questions:

**i)** On which intervals is the graph of `f(x)`

concave up? On which intervals is the graph concave down?

**ii)** List the x-coordinates of all the points of inflection.

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

, assuming that `f(0) = 2`

.

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0.56mins

Q5b

Describe how you would use the second derivative to determine a local minimum or maximum.

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0.51mins

Q6

For the following function,

**i)** determine any points of inflection

**ii)** use the results of part i, along with the revised algorithm, to sketch each function.

`f(x) = x^4 + 4x^3`

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3.07mins

Q8a

For the following function,

**i)** determine any points of inflection

**ii)** use the results of part i, along with the revised algorithm, to sketch each function.

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

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8.02mins

Q8b

Sketch the graph of a function with the following properties:

`f'(x) > 0`

when `x < 2`

and when `2 < x < 5`

`f'(x) < 0`

when `x > 5`

`f'(2) =0`

when `f'(5) = 0`

`f''(x) < 0`

when `x < 2`

and when `4 < x < 7`

`f''(x) > 0`

when `2 < x < 4`

and when `x > 7`

`f(0) = -4`

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2.13mins

Q9

Find constants `a, b`

, and `c`

such that the function `f(x) = ax^3 +bx^2 +c`

will have a local extremum at `(2, 11)`

and a point of inflection at `(1, 5)`

.

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3.46mins

Q10

Find the value of the constant `b`

such that the function `f(x) = \sqrt{x + 1} + \displaystyle{\frac{b}{x}}`

has a point of inflection at `x =3`

.

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2.18mins

Q11

Show that the graph of `f(x) = ax^4 + bx^3`

has two points of inflation. Show that the x-coordinate of one of these points lies midway between the x-intercepts.

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1.33mins

Q12

Use the algorithm for curve sketching to sketch the function

`y = \displaystyle{\frac{x^3 -2x^2 + 4x}{x^2 - 4}}`

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11.55mins

Q13

Find the inflection points, if any exist, for the graph of `f(x) = (x -c)^n`

, for `n= 1, 2, 3`

, and `4`

. What conclusion can you draw about the value of `n`

and the existence of inflection points on the graph of `f`

?

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1.39mins

Q14

Lectures
3 Videos

Introduction to Concavity of a Curve

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6.15mins

Introduction to Concavity of a Curve

Example of Second Derivative Test

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4.30mins

Example of Second Derivative Test

Region of Postivie Concavity example

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2.39mins

Region of Postivie Concavity example