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Solutions
76 Videos

Using the limit definition of the slope of a tangent, determine that slope of the tangent to each curve at the given point.

`y= 3x^2 + 4x - 5, (2, 15)`

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2.25mins

Q1a

Using the limit definition of the slope of a tangent, determine that slope of the tangent to each curve at the given point.

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

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1.44mins

Q1b

Using the limit definition of the slope of a tangent, determine that slope of the tangent to each curve at the given point.

`f(x) = \sqrt{x + 3}, (6 ,3)`

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1.46mins

Q1c

`f(x) = 2^{5x}, (1, 32) `

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3.49mins

Q1d

The position, in metres, of an object is given by `s(t) = 2t^2 +3t + 1`

, where `t`

is the time in seconds.

Determine the average velocity from `t = 1`

to `t = 4`

.

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0.59mins

Q2a

The position, in metres, of an object is given by `s(t) = 2t^2 +3t + 1`

, where `t`

is the time in seconds.

- Determine the the instantaneous velocity at
`t = 3`

.

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0.21mins

Q2b

If `\displaystyle \lim_{h\to 0} \displaystyle{\frac{(4 + h)^3 - 64}{h}}`

represents the slope of the tangent to `y = f(x)`

at `x = 4`

, what is the equation of `f(x)`

?

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0.51mins

Q3

An object is dropped from the observation deck of the Skyline Tower in Niagara Falls, Ontario. The distance, in metres, from the deck at `t`

seconds is given by `d(t) = 4.9t^2.`

- Determine the average rate of change in distance with respect to time from
`t =1`

to`t= 3`

.

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0.39mins

Q4a

An object is dropped from the observation deck of the Skyline Tower in Niagara Falls, Ontario. The distance, in metres, from the deck at `t`

seconds is given by `d(t) = 4.9t^2.`

Determine the instantaneous rate of change in distance with respect to time at 2 s.

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0.32mins

Q4b

An object is dropped from the observation deck of the Skyline Tower in Niagara Falls, Ontario. The distance, in metres, from the deck at `t`

seconds is given by `d(t) = 4.9t^2.`

- The height of the observation deck is 146.9 m. How fast is the object moving when it hits the ground?

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0.58mins

Q4c

The model `P(t) = 2t^2 + 3t + 1`

estimates the population of fish in a reservoir, where `P`

represents the population, in thousands, and `t`

is the number of years since 2000.

Determine the average rate of population change between 2000 and 2008.

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0.44mins

Q5a

The model `P(t) = 2t^2 + 3t + 1`

estimates the population of fish in a reservoir, where `P`

represents the population, in thousands, and `t`

is the number of years since 2000.

Estimate the rate at which the population was changing at the start of 2005.

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0.28mins

Q5b

Given the graph of `f(x)`

at the left, determine the following:

**i)** `f(2)`

**ii)** `\lim_{x\to 2^-}f(x)`

**iii)** `\lim_{x\to 2^+}f(x)`

**iv)** `\lim_{x\to 6}f(x)`

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0.37mins

Q6a

Does `\lim_{x\to 4}f(x)`

exist? Justify your answer.

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0.28mins

Q6b

Consider the following function:
`f(x) =`

`x^2 + 1`

, if `x < 2`

`2x + 1`

, if `x = 2`

`-x+ 5`

, if `x > 2`

Determine where `f(x)`

is discontinuous, and justify your answer.

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1.15mins

Q7

Use algebraic method to evaluate each limit (if it exists).

`\displaystyle \lim_{x\to 0}\frac{2x^2 + 1}{x-5}`

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0.15mins

Q8a

Use algebraic method to evaluate each limit (if it exists).

`\displaystyle \lim_{x\to 3}\frac{x - 3}{\sqrt{x+ 6} - 3}`

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0.51mins

Q8b

Use algebraic method to evaluate each limit (if it exists).

`\displaystyle \lim_{x\to -3} \frac{\frac{1}{x} + \frac{1}{3}}{x + 3}`

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0.33mins

Q8c

Use algebraic method to evaluate each limit (if it exists).

`\displaystyle \lim_{x\to 2} \frac{x^2 - 4}{x^2 - x - 2}`

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0.24mins

Q8d

Use algebraic method to evaluate each limit (if it exists).

`\displaystyle \lim_{x\to 2} \frac{x - 2}{x^3 - 8}`

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0.40mins

Q8e

Determine the derivative of each function from first principles.

`\displaystyle f(x) =3x^2 + x +1`

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1.37mins

Q9a

Determine the derivative of each function from first principles.

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

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0.57mins

Q9b

Determine the derivative of each function.

`\displaystyle y = x^3 -4x^2 + 5x + 2`

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0.43mins

Q10a

Determine the derivative of each function.

`\displaystyle y = \sqrt{2x^3 + 1}`

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0.41mins

Q10b

Determine the derivative of each function.

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

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0.34mins

Q10c

Determine the derivative of the function.

`\displaystyle y = (x^2 +3)^2(4x^5 + 5x + 1)`

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1.04mins

Q10d

Determine the derivative of each function.

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

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0.48mins

Q10e

Determine the derivative of each function.

`\displaystyle y = [x^2 + (2x + 1)^3]^5`

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0.36mins

Q10f

Determine the equation of the tangent to `y = \displaystyle{\frac{18}{(x +2)^2}}`

at the point `(1,2 )`

.

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1.10mins

Q11

Determine the slope of the tangent to `y = x^2 + 9x + 9`

at the point where the curve intersects the line `y = 3x`

.

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0.38mins

Q12

In 1980, the population of Littletown, Ontario, was 1100. After a time `t`

, in years, the population was given by `p(t) =2t^2 + 6t + 1100`

.

Determine `p'(t)`

, the function that describes the rate of change of the population at time `t`

.

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0.17mins

Q13a

In 1980, the population of Littletown, Ontario, was 1100. After a time `t`

, in years, the population was given by `p(t) =2t^2 + 6t + 1100`

.

- Determine the rate of change of the population at the start of 1990.

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0.21mins

Q13b

In 1980, the population of Littletown, Ontario, was 1100. After a time `t`

, in years, the population was given by `p(t) =2t^2 + 6t + 1100`

.

- At the beginning of what year was the rate of change of the population 110 people per year?

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0.33mins

Q13c

Determine `f'`

and `f''`

`f(x) = x^5 -5x^3 + x + 12`

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0.20mins

Q14a

Determine `f'`

and `f''`

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

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0.30mins

Q14b

Determine `f'`

and `f''`

`f(x) = \displaystyle{\frac{4}{\sqrt{x}}}`

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0.45mins

Q14c

Determine `f'`

and `f''`

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

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0.36mins

Q14d

Determine the extreme values of each function on the given interval.

`f(x) = 1 + (x + 3)^2, -2\leq x \leq 6`

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0.44mins

Q15a

Determine the extreme values of each function on the given interval.

`f(x) = x + \displaystyle{\frac{1}{\sqrt{x}}}, 1\leq x \leq 9`

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1.15mins

Q15b

Determine the extreme values of each function on the given interval.

`f(x) = \displaystyle{\frac{e^x}{1 + e^x}}, x \in [0, 4]`

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0.53mins

Q15c

Determine the extreme values of each function on the given interval.

`f(x) = 2\sin 4x + 3, x \in [0, \pi]`

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2.18mins

Q15d

The position, at time `t`

, in seconds, of an object moving along a line is given by `s(t) =3t^3 - 40.5t^2 + 162t`

for `0\leq t \leq 8`

.

- Determine the velocity and the acceleration at any time
`t`

.

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0.16mins

Q16a

The position, at time `t`

, in seconds, of an object moving along a line is given by `s(t) =3t^3 - 40.5t^2 + 162t`

for `0\leq t \leq 8`

.

- When is the object stationary? When is it advancing? When is it retreating?

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1.45mins

Q16b

The position, at time `t`

, in seconds, of an object moving along a line is given by `s(t) =3t^3 - 40.5t^2 + 162t`

for `0\leq t \leq 8`

.

At what time, `t`

, is the velocity not changing?

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1.17mins

Q16c

`t`

, in seconds, of an object moving along a line is given by `s(t) =3t^3 - 40.5t^2 + 162t`

for `0\leq t \leq 8`

.

a) At what time, `t`

, is the velocity decreasing?

b) At what time, `t`

, is the velocity increasing?

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2.53mins

Q16de

A farmer has 750 m of fencing. The farmer wants to enclose a rectangular area on all four sides, and then divide it into four pens of equal size with the fencing parallel to one side of the rectangle. What is the largest possible area of each of the four pens?

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2.41mins

Q17

A cylindrical metal can is made to hold 500 mL of soup. Determine the dimensions of the can that will minimize the amount of metal required. (Assume that the top and sides of the can are made from metal of the same thickness.)

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2.39mins

Q18

A cylindrical container, with a volume of 4000 cm`^3`

, is being constructed to hold candies. The cost of the base and lid is $0.005 cm`^2`

, and the cost of the side walls is $0.0025 cm`^2`

. Determine the dimensions of the cheapest possible container.

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2.54mins

Q19

An open rectangular box has a square base, with each side measuring `x`

centimetres.

If the length, width, and depth have a sum of 140 cm, find the depth in terms of `x`

.

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0.44mins

Q20a

An open rectangular box has a square base, with each side measuring `x`

centimetres.

Determine the maximum possible volume you could have when constructing a box with these specifications.

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1.56mins

Q20b

The price of `x`

MP2 players is `p(x) = 50-x^2`

, where `x \in \mathbb{N}`

. IF the total revenue, `R(x)`

, is given by `R(x) =xp(x)`

, determine the value of `x`

that corresponds to the maximum possible total revenue.

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1.43mins

Q21

An express railroad train between two cities carries 10 000 passengers per year for a one-way fare of $50. If the fare goes up, ridership will decrease because more people will drive. It is estimated that each $10 increase in the fare will result in 1000 fewer passengers per year. What fare will maximize revenue?

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2.26mins

Q22

A travel agent currently has 80 people signed up for a tour. The price of
a ticket is `\$5000`

per person. The agency has chartered a plane seating 150 people at a cost of `\$250 000`

. Additional costs to the agency are incidental fees of `\$300`

per person. For each `\$30`

that the price is lowered, one new person will sign up. How much should the price per person be lowered to maximize the profit for the agency?

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4.43mins

Q23

For the function,

- i. determine the derivative,
- ii. all the critical numbers, and
- iii. the intervals of increase and decrease.

`y = -5x^2 + 20x + 2`

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0.39mins

Q24a

For the function,

- i. determine the derivative,
- ii. all the critical numbers, and
- iii. the intervals of increase and decrease.

`y = 6x^2 + 16x -40`

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0.41mins

Q24b

For the function,

- i. determine the derivative,
- ii. all the critical numbers, and
- iii. the intervals of increase and decrease.

`y = 2x^3 -24x`

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0.46mins

Q24c

For the function,

- i. determine the derivative,
- ii. all the critical numbers, and
- iii. the intervals of increase and decrease.

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

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0.45mins

Q24d

For the of the following,

- i. determine the equations of any horizontal, vertical, or oblique asymptotes and
- ii. all local extrema:

`y =\frac{8}{x^2 - 9}`

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1.24mins

Q25a

For the of the following,

- i. determine the equations of any horizontal, vertical, or oblique asymptotes and
- ii. all local extrema:

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

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5.04mins

Q25b

Use the algorithm for curve sketching to sketch the graph of each function.

`f(x) =4x^3 + 6x^2 -24x -2`

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2.24mins

Q26a

Use the algorithm for curve sketching to sketch the graph of each function.

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

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5.36mins

Q26b

Determine the derivative of each function.

`f(x) = (-4)e^{5x + 1}`

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0.18mins

Q27a

Determine the derivative of each function.

`f(x) = xe^{3x}`

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0.23mins

Q27b

Determine the derivative of each function.

`f(x) = 6^{3x - 8}`

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0.14mins

Q27c

Determine the derivative of each function.

`f(x) = e^{\sin x}`

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0.13mins

Q27d

Determine the equation of the tangent to the curve `y =e^{2x - 1}`

at `x = 1`

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0.50mins

Q28

In a research laboratory, a dish of bacteria is infected with a particular disease. The equation `N(d) =(15d)e^{-\frac{d}{5}}`

models the number of bacteria, `N`

, that will be infected after `d`

days.

- How many days will pass before the maximum number of bacteria will be infected?

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1.01mins

Q29a

In a research laboratory, a dish of bacteria is infected with a particular disease. The equation `N(d) =(15d)e^{-\frac{d}{5}}`

models the number of bacteria, `N`

, that will be infected after `d`

days.

- Determine the maximum number of bacteria that will be infected.

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0.20mins

Q29b

Determine the derivative of each function.

`y = 2\sin x -3\cos 5x`

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0.26mins

Q30a

Determine the derivative of each function.

`y = (\sin 3x + 1)^4`

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0.28mins

Q30b

Determine the derivative of each function.

`y = \sqrt{x^2 + \sin 3x}`

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0.25mins

Q30c

Determine the derivative of each function.

`y = \frac{\sin x}{\cos x + 2}`

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0.43mins

Q30d

Determine the derivative of each function.

`y =\tan x^2 - \tan^2 x`

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0.34mins

Q30e

Determine the derivative of each function.

`y = \sin(\cos x^2)`

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0.38mins

Q30f

A tool shed, `250 cm`

high and `100 cm`

deep, is built against a wall. Calculate the shortest ladder that can reach from the ground, over the shed, to the wall behind.

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4.58mins

Q31

A corridor that is 3 m wide makes a right-angle turn, as shown on the left. Find the longest rod that can be carried horizontally around this corner. Round your answer to the nearest tenth of a metre.

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3.13mins

Q32