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Solutions
29 Videos

Each graph represents the rate of change of a function. Determine a possible equation for the function.

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Q1a

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Q1b

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Q1c

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Q1d

a) Describe one method that you could use to estimate the value of e, assuming you do not know its value, and without using the "e" button on a calculator. Use only the following definition:

` e = \lim_{ n \to \infty}(1+ \dfrac{1}{n})^n, n \in \mathbb{N}`

b) Use your method to estimate the value of e, correct to two decimal places.

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Q2

a) Graph the function ` y = e^{-x}`

b) Graph the inverse of this function by reflecting the curve in the line ` y = x`

.

c) What is the equation of this inverse function? Explain how you know.

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Q3

Evaluate, rounding to three decimal places, if necessary.

a) `e^{-3}`

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Q4a

Evaluate, rounding to three decimal places, if necessary.

a) `ln(6.2)`

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Q4b

Evaluate, rounding to three decimal places, if necessary.

a) `ln(e^{\dfrac{3}{4}})`

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Q4c

Evaluate, rounding to three decimal places, if necessary.

a) `e^{ln(0.61)}`

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Q4d

Solve for `x`

, correct to two decimal places.

`9 = 3e^x`

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Q5a

Solve for `x`

, correct to two decimal places.

` lnx= -5`

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Q5b

Solve for `x`

, correct to two decimal places.

` x = 10e ^{\dfrac{-3}{2}}`

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Q5c

Solve for `x`

, correct to two decimal places.

` 10 = 100e ^{\dfrac{-x}{4}}`

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Q5d

The population of a bacterial culture as a function of time is given by the equation `P(t) = 50e^{0.12t}`

, where `P`

is the population after `t`

days.

a) What is the initial population of the bacterial culture?

b) Estimate the population after 4 days.

c) How long will it take for the population to double?

d) Rewrite `P(t)`

as an exponential having base 2.

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Q6

a) Differentiate each function with respect to `x`

.

i) `f(x) = (\dfrac{1}{2})^x`

ii) `g(x) = -2e^x`

b) Graph each function from part a) and its derivative on the same grid.

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Q7

Find the equation of the line that is tangent to the curve `y= 2(3^x)`

at ` x= 1`

.

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Q8

Find the equation of the line that is tangent to the curve `y = -3e^x`

at ` x = ln2`

.

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Q9

An investor places $1000 into an account whose value increases according to the function `A(x) = 1000(2)^{\dfrac{t}{9}}`

, where `A`

is the investment's value after `t`

years.

a) Determine the value in the account after 5 years.

b) How long will it take for this investment to

i) double in value?

ii) triple in value?

c) Find the rate at which the investment is growing at each of these times.

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Q10

Determine the derivative of each function.

` y = e^{3x^2 - 2x +1}`

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Q11a

Determine the derivative of each function.

` f(x) = (x-1)e^{2x}`

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Q11b

Determine the derivative of each function.

` y = 3x + e^{-x}`

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Q11c

Determine the derivative of each function.

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

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Q11d

Determine the derivative of each function.

` g(x) = (\dfrac {1}{3})^{4x} - 2e^{\sin x}`

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Q11e

Identify the coordinates of any local extrema of the function ` y = e^{x^2}`

and classify each as either a local maximum or a local minimum.

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Q12

Identify the coordinates of any local extrema of the function `y = 2e^x`

and classify each as either a local maximum or a local minimum.

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Q13

A certain computer that is purchased today depreciates in value according to the function ` V(t) = 900e6{\dfrac{-t}{3}}`

, where `t`

represents times, in years.

a) What was the purchase price of the computer?

b) What is its value after 1 year?

c) How long will it take for the computer's value to decrease to half of its original value?

d) At what rate will the computer's value be depreciating at this time?

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Q14

A 80-mg sample of protactinium-233 (Pa-223) is placed in a nuclear reactor. After 5 days, the sample has decayed to 70 mg. The amount of Pa-223 remaining in the reactor can be modelled by the function:

` N(t) = N_0 (\dfrac{1}{2}) ^{\dfrac{t}{h}}`

Here, `N(t)`

represents the amount of Pa-223, in milligrams, as a function of time, `t`

, in days; `N_0`

represents the initial amount of Pa-223, in milligrams; and `h`

represents the half-life of Pa-223, in days.

a) Determine the half-life of Pa-223.

b) Write an equation that gives the amount of Pa-223 remaining as a function of time, in terms of its half-life.

c) How fast is the sample decaying after 5 days?

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Q15

After Lee gives his little sister Kara a big push on a swing, her horizontal position as a function of time is given by the equation `x(t) = 3\cos t(e ^{-0.05t})`

where `x(t)`

is her horizontal displacement, in metres, from the lowest point of her swing, as a function of time, `t`

, in seconds.

a) From what horizontal distance from the bottom of Kara's swing did Lee push his sister?

b) Determine the greatest speed Kara will reach and when this occurs.

c) How long will it take for Kara's maximum horizontal displacement at the top of her swing arc to diminish to 1 m? After how many swings will this occur?

d) Sketch the graph of this function.

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Q16