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Lectures
4 Videos

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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

Sketching cubic root function

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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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