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Solutions
49 Videos

Determine `f'`

and `f''`

, if `f(x)=x^4-\displaystyle{\frac{1}{x^4}}`

.

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0.39mins

Q1

For `y=x^9-7x^3+2`

, find `\displaystyle{\frac{d^2y}{dx^2}}`

.

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0.24mins

Q2

Determine the velocity and acceleration of an object that movies along a straight line in such a sway that its position is `s(t)=t^2+(2t-3)^{\displaystyle{\frac{1}{2}}}`

.

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0.51mins

Q3

Determine the velocity and acceleration as functions of time, `t`

, for `s(t)=t-7+\displaystyle{\frac{5}{t}}`

, `t\neq0`

.

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0.32mins

Q4

A pellet is shot into the air. Its height above the ground at any time, `t`

, is defined by `s(t)=45t-5t^2`

. For what values of `t`

, `t\geq0`

, is the upward velocity of the pellet positive? For what values of `t`

is the upward velocity zero and negative?

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0.34mins

Q5

Determine the maximum and minimum of each function on the given interval.

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

, `-2\leq x \leq 4`

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1.21mins

Q6a

Determine the maximum and minimum of each function on the given interval.

`f(x)=12x-x^3`

, `x\in[-3,5]`

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0.38mins

Q6b

Determine the maximum and minimum of each function on the given interval.

`f(x)=2x+\displaystyle{\frac{18}{x}}`

, `1\leq x \leq 5`

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0.59mins

Q6c

A motorist starts breaking for a stop sign. After `t`

seconds, the distance, in metres, from the front of the car to the sign is `s(t)=62-16t+t^2`

.

How far was the front of the car from the sign when the driver started braking?

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0.09mins

Q7a

A motorist starts breaking for a stop sign. After `t`

seconds, the distance, in metres, from the front of the car to the sign is `s(t)=62-16t+t^2`

.

Does the car go beyond the stop sign before stopping?

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1.04mins

Q7b

The position function of an object that moves in a straight line is `s(t)=1+2t-\displaystyle{\frac{8}{t^2+1}}`

, `0 \leq t \leq 2`

. Calculate the maximum and minimum velocities of the object over the given time interval.

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3.17mins

Q8

Suppose that the cost, in dollars, of manufacturing `x`

items is approximated by `C(x)=625+15x+0.01x^2`

, for `1\leq x \leq 500`

. The unit cost (the cost of manufacturing one item) would then be `U(x)=\displaystyle{\frac{C(x)}{x}}`

. How many items should be manufactured to ensure that the unit cost is minimized?

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3.19mins

Q9

For each of the following cost functions, determine

i. the cost of producing 400 items

ii. the average cost of each of the first 400 items produced

iii. the marginal cost when `x=400`

, as well as the cost of producing the 401st item

`C(x)=3x+1000`

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1.32mins

Q10a

The position of an object moving along a straight line is described by the function `s(t)=3t^2-10`

for `t\geq 0`

. Is the object moving toward or away from its starting position when `t=3`

?

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0.29mins

Q12a

The position of an object moving along a straight line is described by the function `s(t)=-t^3+4t^2-10`

for `t\geq 0`

. Is the object moving toward or away from its starting position when `t=3`

?

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0.27mins

Q12b

A particle moving along a straight line will be `s`

centimetres from a fixed point at time `t`

seconds, where `t>0`

and `s=27t^3+\displaystyle{\frac{16}{t}}+10`

.

Determine when the velocity will be zero.

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0.52mins

Q13a

A particle moving along a straight line will be `s`

centimetres from a fixed point at time `t`

seconds, where `t>0`

and `s=27t^3+\displaystyle{\frac{16}{t}}+10`

.

Is the particle accelerating? Explain.

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0.49mins

Q13b

A box with a square base and no top must have a volume of 10000 cm`^3`

. If the smallest dimension is 5 cm, determine the dimensions of the box that minimize the amount of material used.

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3.45mins

Q14

An animal breeder wishes to create five adjacent rectangular pens, each with an area of 2400 m`^2`

. To ensure that the pens are large enough for grazing, the minimum for either dimension must be 10 m. Find the dimensions required for the pens to keep the amount of fencing used to a minimum.

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3.15mins

Q15

You are given a piece of sheet metal that is twice as long as it is wide and has an area of 800 m`^2`

. Find the dimensions of the rectangular box that would contain a maximum volume if it were constructed from this piece of metal by cutting out squares of equal area at all four corners and folding up the sides. The box will not have a lid. Give your answer correct to one decimal place.

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4.45mins

Q16

A cylindrical can needs to hold 500 cm`^3`

of apple juice. The height of the can must be between 6 cm and 15 cm, inclusive. How should the can be constructed so that a minimum amount of material will be used in the construction? (Assume that there will be no waste.)

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4.22mins

Q17

In oil pipeline construction, the cost of pipe to go underwater is 60% more than the cost of pipe used in dry-land situations. A pipeline comes to a river that is 1 km wide at point A and must be extended to a refinery, `R`

, on the other side, 8 km down the river. Find the best way to cross the river (assuming it is straight) so that the total cost of the pipe is kept to a minimum. (Give your answer correct to one decimal place.)

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4.30mins

Q18

A train leaves the station at 10:00 p.m. and travels due north at a speed of 100 km/h. Another train has been heading due west at 120 km/h and reaches the same station at 11:00 p.m. At what time were the two trains closest together?

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4.32mins

Q19

A store sells portable MP3 players for $100 each and, at this price, sells 120 MP3 players every month. The owner of the store wishes to increase his profit, and he estimate that, for every $2 increase in the piece of MP3 players, one less MP3 player will be sold each month. If each MP3 player costs the store $70, at what price should the store sell the MP3 players to maximize profit?

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2.41mins

Q20

An offshore oil well, `P`

, is located in the ocean 5 km form the nearest point on the shore, `A`

. A pipeline is to be built to take oil from `P`

to a refinery that is 20 km along the straight shoreline from `A`

. If it costs $100000 per kilometre to lay pipe underwater and only $75000 per kilometre to lay pipe on land what route from the well to the refinery will be the cheapest? (Give your answer correct to one decimal place.)

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4.59mins

Q21

The printed area of a page in a book will be 81 cm`^3`

. The margins at the top and bottom of the page will each be 3 cm deep. The margins at the sides of the pages will each be 2 cm wide. What page dimensions will minimize the amount of paper?

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2.43mins

Q22

A rectangular rose garden will be surrounded by a brick wall on three sides and by a fence on the fourth side. The area of the garden will be 1000 m`^2`

. The cost of the brick wall is $192/m. The cost of the fencing is $48/m. Find the dimensions of the garden so that the cost of the material will be as low as possible.

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3.26mins

Q23

A boat leaves a dock at 2:00 p.m., heading west at 15 km/h. Another boat heads south at 12 km/h and reaches the same dock at 3:00 p.m. When were the boats closest to each other?

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3.20mins

Q24

Two towns, Ancaster and Dundas, are 4 km and 6 km, respectively, from an old railroad line that has been made into a bike trail. Points `C`

and `D`

on the trail are the closest points to the two towns, respectively. These points are 8 km apart. Where should a rest stop be built to minimize the length of new trail that must be built from both town to the rest stop?

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5.01mins

Q25

Find the absolute maximum and minimum values.

`f(x)=x^2-2x+6`

, `-1\leq x\leq 7`

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0.48mins

Q26a

Find the absolute maximum and minimum values.

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

, `-3\leq x \leq 3`

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0.42mins

Q26b

Find the absolute maximum and minimum values.

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

, `-5\leq x\leq 5`

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0.40mins

Q26c

Find the absolute maximum and minimum values.

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

, `-2\leq x\leq 4`

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0.49mins

Q26d

Sam applies the brakes steadily to stop his car, which is travelling at 20 m/s. The position of the car, `s`

, in metres at `t`

seconds, given by `s(t)=20t-0.3t^3`

. Determine

the stopping distance

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0.54mins

Q27a

Sam applies the brakes steadily to stop his car, which is travelling at 20 m/s. The position of the car, `s`

, in metres at `t`

seconds, given by `s(t)=20t-0.3t^3`

. Determine

- the stopping time

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1.05mins

Q27b

Sam applies the brakes steadily to stop his car, which is travelling at 20 m/s. The position of the car, `s`

, in metres at `t`

seconds, given by `s(t)=20t-0.3t^3`

. Determine

the acceleration at t =2s

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0.49mins

Q27c

Calculate each of the following:

`f''(2)`

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

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0.18mins

Q28a

Calculate each of the following:

`f''(-1)`

if `f(x)=-2x^{-3}+x^2`

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0.30mins

Q28b

Calculate each of the following:

`f''(0)`

if `f(x)=(4x-1)^4`

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0.47mins

Q28c

Calculate each of the following:

`f''(1)`

if `f(x)=\displaystyle{\frac{2x}{x-5}}`

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1.00mins

Q28d

Calculate each of the following:

`f''(4)`

if `f(x)=\sqrt{x+5}`

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1.03mins

Q28e

Calculate each of the following:

`f''(8)`

if `f(x)=\sqrt[3]{x^2}`

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0.53mins

Q28f

An object moves along a straight line. The object's position at time `t`

is given by `s(t)`

. Find the position, velocity, acceleration, and speed at the specified time.

`s(t)=\displaystyle{\frac{2t}{t+3}}`

, `t=3`

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1.17mins

Q29a

An object moves along a straight line. The object's position at time `t`

is given by `s(t)`

. Find the position, velocity, acceleration, and speed at the specified time.

`s(t)=t+\displaystyle{\frac{5}{t+2}}`

, `t=1`

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1.17mins

Q29b

The function `s(t)=(t^2+t)^{\displaystyle{\frac{2}{3}}}`

, `t \geq 0`

, represents the displacement, `s`

, in metres of a particle moving along a straight line after `t`

seconds.

Determine `v(t)`

and `a(t)`

.

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2.51mins

Q30a

The function `s(t)=(t^2+t)^{\displaystyle{\frac{2}{3}}}`

, `t \geq 0`

, represents the displacement, `s`

, in metres of a particle moving along a straight line after `t`

seconds.

Determine `v(t)`

and `a(t)`

.

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0.34mins

Q30b

The function `s(t)=(t^2+t)^{\displaystyle{\frac{2}{3}}}`

, `t \geq 0`

, represents the displacement, `s`

, in metres of a particle moving along a straight line after `t`

seconds.

Find the average velocity during the first 5 s.

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0.32mins

Q30c

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

, `t \geq 0`

, represents the displacement, `s`

, in metres of a particle moving along a straight line after `t`

seconds.

Find the average acceleration during the first 5 s.

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0.29mins

Q30d

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

, `t \geq 0`

, represents the displacement, `s`

, in metres of a particle moving along a straight line after `t`

seconds.

Determine the acceleration at exactly 5 s.

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Q30e