For each of the following functions:
Find any stationary points of the function in the given domain.
Find the absolute maximum value of the function in the given domain.
Find the absolute minimum value of the function in the given domain.
f: \left[ - 7 , 5\right] \to \Reals given by f \left( x \right) = - 6 x^{2} - 12 x + 90.
f: \left[3, 8\right] \to \Reals given by f \left( x \right) = 2 x^{2} - 2 x - 24.
f: \left[ - 7 , 3\right] \to \Reals given by f \left( x \right) = 8 x^{2} + 32 x - 96.
f: \left[ - 7 , - 4 \right] \to \Reals given by f \left( x \right) = - 4 x^{2} - 8 x + 60.
f: \left[4, 14\right] \to \Reals given by f \left( x \right) = x^{3} - 19 x^{2} + 104 x - 137.
Consider the function f \left( x \right) = 2 x - 4 \sqrt{x}.
Find the stationary point of the function.
Find the absolute minimum value of the function within the range \dfrac{1}{2} \leq x \leq 2.
If A = x y and 20 x + 5 y = 160:
Find an expression for A in terms of x only.
Determine the possible values of x that maximise A.
Find an expression for the second derivative, A''.
Do the values of x found in part (b) maximise A? Explain your answer.
State the maximum value of A.
Calculate the value of y that maximises A.
A potato farmer finds that the yield per square metre, y, when spacing his plants, x (between 0.5 \text{ m} and 3.0 \text{ m}), approximates the following equation: y = - \dfrac{x^{2}}{4} + \dfrac{7 x}{20}
Find \dfrac{d y}{d x}.
For what value of x is \dfrac{d y}{d x} equal to 0?
Find the maximum possible yield. Round your answer to two decimal places.
A parabolic satellite dish is pointing straight up. Along a cross-section that passes through the centre of the dish, the height in metres above ground is given by the following equation:
y = \dfrac{1}{100} \left(x^{2} - 100 x\right) + 50
Find \dfrac{d y}{d x}.
For what value of x is \dfrac{d y}{d x} equal to 0?
How far above the ground is the dish at its lowest point?
A passenger's height in metres along a section of roller coaster can be described by the function: H(x)=15 \cos (\dfrac{\pi x}{50}) +20 for 0 \lt x \leq 100, where the passenger is x metres from the start of the given section of the roller coaster.
Find the gradient function.
What is the position on the section that has the steepest downwards gradient?
Would this point be represented by a minimum or maximum point on the gradient function?
What is the position on the section that has the steepest uphill gradient?
Would this point be represented by a minimum or maximum point on the gradient function?
A population P, in thousands of insects, was observed over the course of a year to follow the model: P(t)= 2.5 \sin (\dfrac{\pi t}{6}) +4 for 0 \leq t \leq 12, with time t measured in months after January 1st.
Find P'(t).
In what month is the population at its highest?
In what month is the population increasing at the fastest rate?
What is the fastest rate of decrease in the population over the year?
Charlie is fencing off a rectangular section of his backyard to use as a vegetable garden. He uses the existing back wall as one of the longer sides and has 24 \text{ m} of fencing to create the other three sides. He wishes to make the area for vegetables as large as possible.
Write an equation for the length, L, of the vegetable garden in terms of the width, W.
Hence write an equation for the area, A, of the vegetable garden in terms of the width, W.
Find the width that could maximise the area of the vegetable garden.
Find A'', for this width.
Does the width found in part (c) give a maximum area? Explain your answer.
Find the length of the vegetable garden that maximises the area.
Find the maximum area of the vegetable garden.
A box without a lid is to be constructed from a piece of rectangular cardboard that measures 90 \text{ cm} by 42 \text{ cm}. Four identical squares will be cut out of the corners of the rectangle to allow the sides to fold up:
Let x be the height of the box, and V be the volume of the box.
Form an equation for V, in terms of x.
Find the value of x that could correspond to the box of largest volume.
Show that the volume of the box is maximised at this x-value.
Calculate the maximum volume of the box.
A rectangular sheet of cardboard measuring 96 \text{ cm} by 60 \text{ cm} is to be used to make an open box. A square of width w \text{ cm} is removed from each corner to make the net shown:
Write an expression for the area, A, of the base of the box in terms of w.
Write an expression for the volume, V, of the box in terms of w.
Find the possible value(s) of w that could maximise the volume of the box.
Find an expression for the second derivative, V''.
Find the value of w that maximises the volume of the box.
Hence determine the dimensions of the box with the maximum volume.
A rectangular box is to be made with the following constraints:
- The length must be 5 times the width.
- The total length of the edges must be 144 \text{ m}.
Write an equation for the height of the box, h, in terms of the width, w.
Write an expression for the volume, V, of the box in terms of the width, w.
Find the value of w that might maximise the volume of the box.
Find an expression for the second derivative, V''.
Show that the volume of the box is maximised for the value of w found in part (c).
Calculate the maximum volume of the box.
Hence state the dimensions of the box with the maximum volume.
When exam results were released online, the following formula was used to approximate the number of students, M, logged onto the site at any time over the first 8.5 hours:
M \left( t \right) = 324 + 77 t^{2} - t^{4}Initially, at t=0, how many students logged on to to check their results?
Find the whole number of students who logged on to view their results at the end of the 8.5 hour time period.
Find the exact values of t for which M' \left( t \right) = 0.
Find the maximum whole number of students logged onto the website at any one time in the first 8.5 hours.
Find the exact values of t at which the students were logging on to the site most rapidly.
A cylinder has a height of h\text{ cm} and a radius that is 90\text{ cm} less the height.
Write an expression for the radius in terms of h.
Write an expression for the volume of the cylinder, V, in terms of h.
Find the first derivative, V'.
Find the possible two values of h that maximise the volume of the cylinder.
Find the second derivative, V''.
Find the value of h for which the volume of the cylinder is a maximum.
Hence, find the maximum volume in exact form.
Find the radius for which the volume of the cylinder is a maximum.
An open rectangular box is to be made with volume 0.8 \text { m}^3. The box will have a square base of side x \text { m}, and a height of h \text { m}. The material used for the sides of the box costs \$ 2 \text{/m}^{2}. The material used for the base, which is to be stronger, costs \$ 6 \text{/m}^{2}.
Use the volume to find an expression for h, in terms of x only.
Hence write an expression for the cost, C, of making the box in terms of x.
Find the possible exact values of x that minimise the cost of making the box.
Find an expression for the second derivative, C''.
Show that a minimum cost occurs for the value of x found in part (c).
Calculate the minimum cost of making the box. Round your answer to the nearest whole dollar.
Calculate the height of the box that achieves this minimum cost, correct to two decimal places.
A rectangular door that is x \text{ m} wide and y \text{ m} tall is to be inscribed into a triangular wall. The wall is isosceles, measuring 4 \text{ m} across the base and 3 \text{ m} high. The door cannot be shorter that 1.8 \text{ m} or narrower than 0.6 \text{ m}.
Write an equation for y in terms of x.
Find an expression for the area, A, of the door in terms of x.
Find \dfrac{d A}{d x}.
Find the minimum possible area of the door.
A rectangular beam of width w \text{ cm} and depth d \text{ cm} can be cut from a cylindrical log of wood as shown in the diagram below. The diameter of the cross-section of the log is \sqrt{27} \text{ cm}. The strength S of the beam is proportional to the product of its width and the square of its depth, so that S = k d^{2} w, where k is a positive constant.
Show that S = k \left( 27 w - w^{3}\right).
Find the value of w that will give a beam of maximum strength.
Find the value of d that will give a beam of maximum strength.
Hence, find the maximum strength of a rectangular beam.
A square beam with diagonal \sqrt{27} \text{ cm} is to be cut from an identical log. Find the strength of this beam.
By what percentage is the maximum strength rectangular beam stronger than the square beam? Round your answer to one decimal place.
A manufacturer makes cylindrical tins of volume 600 \text { cm}^3. The manufacturer wishes to make tins with the smallest possible surface area that still meet the volume requirement.
Use the given volume to write an equation for the height, h, of the cylinder in terms of the radius, r.
Hence write an equation for the surface area, A, of the cylinder in terms of the radius, r.
Find the exact values of r that could minimise the surface area.
Find an expression for the second derivative, A''.
Show that the surface area is minimised for the value of r found in part (c).
Calculate the minimum surface area, correct to the nearest square centimetre.
Calculate the height of the cylinder with the minimum surface area. Round your answer to the nearest centimetre.
A construction company has 2.5 \text{ m}^{3} of concrete to make an access ramp. The ramp is a triangular prism whose triangular faces are right-angled. The height of the ramp is 62.5 \text{ cm}.
Find an equation for the length, l, of the ramp in terms of its width, w.
Find \dfrac{d l}{d w}.
Are there any values of w for which \dfrac{d l}{d w} is equal to 0?
If the width cannot be less than 80 \text{ cm} or greater than 1.6 \text{ m}, find the minimum length of the ramp.
A rectangular cereal box with a square base and open top, is to have a volume of 256 \text { cm}^3.
Find the possible values of x that will minimise the amount of material required to make the box. Round your answers to the nearest integer.
Show that a minimum surface area is achieved for the value of x found in part (a).
A cone is inscribed in a sphere of fixed radius m, centred at O. The height of the cone is x and the radius of the base is r, as shown in the diagram:
Write an expression for the volume, V, of the cone in terms of m and x.
Find the values of x for which the volume of the cone will be a maximum.
Pauline is building a plastic sheet green-house. The plan is to spread the plastic over 4 frames of heavy gauge wire. Each frame will have straight sides and a semicircular roof as shown in the diagram. She can only afford 64 \text{ m} of wire and wants to maximise the capacity of the green-house.
Write an equation for y in terms of x.
Find an expression for the cross-sectional area, A, of the frame in terms of x.
Find \dfrac{d A}{d x}.
For what value of x is\dfrac{d A}{d x} equal to 0? Round your answer to one decimal place.
If the minimum height of the straight sides is 1 \text{ m}, find the maximum value for x. Round your answer to one decimal place.
Find the maximum cross-sectional area of each frame, which would maximise the capacity of the green-house. Round your answer to one decimal place. Round your answer to the nearest whole \text{m}^{3}.
The organisers of a fundraising event are trying to work out what they should charge per ticket to receive the maximum possible revenue. They expect 1200 people if they charge \$11 per ticket. For each \$0.25 drop in price of the ticket, they expect an extra 120 people to attend.
Write an expression for the price of a ticket after x drops of \$0.25 in the cost of a ticket.
Write an expression for the number of people expected to attend after x drops of \$0.25 in the cost of a ticket.
Write an equation for the revenue, R, the organisers can expect to receive, in terms of x.
Find the possible value of x that maximises the revenue.
Find the second derivative, R''.
Does the value of x found in part (d) maximise the revenue? Explain your answer.
Hence find:
The price per ticket that will maximise revenue.
The number of people attending that will maximise revenue.
A cylinder of radius r and height h is to be inscribed in a right circular cone with radius 8 \text{ cm} and height 13 \text{ cm}:
Write an equation for r in terms of h.
Find the exact value of h that results in the cylinder of greatest volume being inscribed in the cone.
The concentration, x, of a certain medication in the bloodstream of a patient, t hours after taking a dose, is given by x = 8 t e^{ - 4 t }.
State the concentration in the bloodstream when the patient initially takes the dose.
Find the time, t, at which the patient has the greatest concentration of medication in his bloodstream.
Find the maximum concentration of the medication. Round your answer to three decimal places.
According to the model, does the concentration ever reach 0 again?
An experimental pesticide is introduced to quickly eradicate the number of pests present in an agricultural region. The population, in thousands, t months after the pesticide is introduced is given by: P \left( t \right) = \dfrac{25 e^{\frac{t + 5}{25}}}{t + 5}
Calculate the initial population of pests, as the pesticide is first introduced. Round your answer to the nearest whole number.
Determine the number of months for the pest population to reach a minimum.
Determine the minimum value of the pest population, to the nearest whole number.
Find the limiting value of P \left( t \right) as t approaches \infty.
State whether the pesticide is effective in the short term.
State whether the pesticide eradicates the pests completely. Explain your answer.
The stock price of a new company can be modelled in the short term by: S \left( t \right) = 8 \cos \left( \dfrac{\pi}{6} t + \dfrac{\pi}{2}\right) + \dfrac{2 \sqrt{3} \pi}{3} t + 14 where S is the stock price in dollars, and t is the number of months since the company floated.
Calculate the initial float price.
Determine the month in which the maximum price is achieved, and the month in which the minimum price is achieved.
Calculate the overall change in the share price between the minimum and maximum share price during the 12 months.
A cylindrical tin can with radius, r \text{ cm}, and height, h \text{ cm}, is to be designed so that the total surface area including the top and bottom is 5 \pi \text { cm}^{2}.
Express h in terms of r.
Let V be the volume of the can. Express V in terms of r.
Find the possible exact value for the radius of the base that could result in the can of the largest volume.
Show that the value of r found in part (c) maximises the volume of the cylinder.
A tunnel is to be built through a mountain so that cargo trucks can service small villages from the nearby city.
Rectangle MNPQ shows the opening of the tunnel. Let x be the distance OM in metres. The shape of the mountain is modelled by the curve:
y = 4 e^{\frac{- x^{2}}{14}}
Let A represent the area of rectangle MNPQ, the cross-section of the opening of the tunnel. Write an equation for A in terms of x.
Find the exact value of x, that maximises the cross-sectional area.
Assuming the cross-section of the tunnel remains uniform from one end to the other, calculate the exact volume of earth that will need to be removed to form the tunnel if it is to be 3.8 \text{ km} long.
In the diagram, the function y = \dfrac{b}{x^{2} + a} has a maximum turning point at \left(0, 4\right) and passes through \left(3, 1\right).
A rectangle BCDE is inscribed within the curve as shown. OY is the axis of symmetry.
Find an expression for b in terms of a using the turning point.
Find the value of a.
Hence, find the value of b.
If C has coordinates \left(x, 0\right), find the \\y-coordinate of B, in terms of x.
Find the area, A(x), of rectangle BCDE, in terms of x.
Determine the exact value of x that results in the maximum area.
Hence, find the maximum area of the rectangle, correct to two decimal places.
Consider the rectangle shown inside a circle of radius 7. The measurements are in centimetres and 0 \leq \theta \leq \dfrac{\pi}{2}.
Find an expression for x in terms of \theta.
Find an expression for y in terms of \theta.
Hence find an expression for the area, A, in terms of \theta.
By differentiating A with respect to \theta, and then using technology, find the possible values of \theta which could maximise the area.
What can we conclude about A when \theta = answer to part (d)?
Hence find the maximum value of the area of the rectangle.
The diagram shows the curve y = 13 - x^{2} and the line y = m, where 0 < m < 13. The line cuts the parabola at points A and B, and C and D are vertically below points A and B respectively on the x-axis:
Let the area of ABCD be represented by R. Write an equation for R in terms of m.
Find an expression for \dfrac{dR}{dm}.
Determine the range of values of m, such that R is increasing.
Find the local maximum function value of R over the domain 0 < m < 13. Round your answer to one decimal place.
Find the largest possible area of ABCD if the values of m are restricted to 8 \leq m \leq 9. Round your answer to one decimal place.
Complete the table to use the first derivative test to show the area of ABCD is a maximum. Round all values to one decimal place.
| m | 8 | 9 | |
|---|---|---|---|
| \dfrac{dR}{dm} | |||
| \text{Sign} | \text{Zero} |
Point A with coordinates \left(p, q\right) lies on the line y = 3 x. Point B has coordinates \left(5, 0\right).
Write an expression for q in terms of p.
Write an equation in terms of p only, for the distance, D, between point A and point B.
Find the values of p that minimise the distance, D.
Find an expression for the second derivative, D''.
Do the values of p found in part (c) minimise the distance, D? Explain your answer.
Find the coordinates of point A.
A car is initially 30 \text { km} due west of a van. The car drives due east at 60 \text { km/h} while the van drives due south at 85 \text { km/h} .
Write an expression for the distance A, travelled by the van from its initial position after t hours.
Write an expression for the distance B, travelled by the car from the initial position of the van after t hours.
Write an expression for the distance D between the vehicles, in terms of t.
Determine the exact number of hours taken for the distance, D, to be a minimum.
Show that the second derivative is given by: D'' = \dfrac{1}{\left( 10\,825 t^{2} - 3\,600 t + 900\right)^{\frac{1}{2}}} \left(10\,825 - \dfrac{\left( 10\,825 t - 1\,800\right)^{2}}{10\,825 t^{2} - 3\,600 t + 900}\right)
Use the second derivative to explain how the time found in part (d) gives a minimum distance.
Calculate the minimum distance between the two vehicles. Round your answer to the nearest whole kilometre.
Towns P, Q and R are located 6 \text{ km} due west, 6 \text{ km} due east and 10 \text{ km} due south respectively, of a town S. A road is to run due north from town R to a point A which is x \text{ km} from R . From A, a branch road is to run to town P and another branch road to town Q.
Find an expression for AS in terms of x.
Find an expression for AP in terms of x.
Let L be the total length of the roads AP, AQ and AR. Express L in terms of x.
Find the exact value of x that results in the minimum total distance of the roads.
Find the exact minimum value of L.
In a new mining town, coal needs to be transported from the mine at point M to a port at point P. There is an existing train line running 100 \text { km} from Q to P such that M is 25 \text { km} from Q. A direct road is to be built that connects point M to point R on the train track. R is x \text { km} from Q.
Mine operators find that transporting the coal by road costs \$130 \text {/km} , and by rail it costs \$100 \text {/km} . Find an expression for the total cost of transportation, C, in terms of x.
Write an expression for the first derivative \dfrac{d C}{d x}.
Find the possible values of x that minimise the cost of transporting the coal, correct to the nearest kilometre.
Complete the table, rounding answers to two decimal places:
| x | 29 | 30 | 31 |
|---|---|---|---|
| \dfrac{dC}{dx} |
Explain the meaning of the values in the table in regards to the cost of transporting the coal.
In a sporting event, competitors start in the water at point B, a buoy 2.5 \text{ km} from the beach. They swim through open water to point C, which is somewhere along the 3 \text{ km} beach between A and D . From there they do a sand run directly to the finish line at point D.
During training, Laura ran at an average speed of 5 \text{ km/h} and swam at an average speed of 2 \text{ km/h}.
Calculate how long would it take Laura if she swam directly from point B to point D. Give your answer in hours, rounded to two decimal places.
Calculate how long it would Laura take if she first swam directly to point A and then ran the whole length of the beach to point D. Give your answer in hours, rounded to two decimal places.
Explain why a combination of swimming and running will help Laura finish the event faster.
Point C is x \text{ km} from point A. If Laura swims to C, and then runs to D, find an expression for the total time, T, in terms of x.
Find the distance, x, which will minimise Laura's time, T, correct to two decimal places.
Hence find the fastest time in which Laura can complete the event, correct to two decimal places.
Lisa finds a tiny burn in her new rectangular scarf, which measures 60\text{ cm} by 420\text{ cm}.
She decides that the best solution is to cut off one corner of the scarf through the burn.
The burn is located 6\text{ cm} from one edge of the scarf and 14\text{ cm} from the end of the scarf:
Find an expression for x in terms of \theta.
Find an expression for y in terms of \theta.
Hence, write an expression for the area of the triangle that Lisa will cut out, A, in terms of \theta.
Find the minimum area of the triangle she has to cut off the scarf to remove the burn. Round your answer to the nearest whole number.
Two corridors meet at a right angle as shown in the diagram. One has a width of 4 metres, and the other has a width of 5 metres.
The angle \theta is the angle between the wall and the diagonal of the corridor.
Write an expression for b in terms of \theta.
Write an expression for x in terms of \theta.
Find L the largest length of a ladder that can be moved around the corner. Round your answer to three decimal places.
A trough for holding water is formed by taking a piece of sheet metal 210\text{ cm} wide and folding 70 cm on either end up as shown below:
Write an expression for y in terms of \theta.
Write an expression for x in terms of \theta.
Hence, write an expression for the area of the cross section of the trough, A, in terms of \theta.
Find the maximum value of the area of the cross section of the trough. Give your answer as an exact value.
A circular lake with a radius of 2\text{ km} s shown below. Georgia swims across the interval x at 2\text{ km/h} and then walks around the arc y at 5\text{ km/h}.
Find an expression for the distance x in terms of \theta.
Find an expression for the distance y in terms of \theta.
Hence, find an expression for T, the total time Georgia spent travelling.
Find the maximum time Georgia can take on the route, correct to three decimal places.
The figure shows a circle with centre O and radius 4\text{ cm}. M and N are points on the circle such that MP \perp PN, PN \perp ON and \angle MON = u, where 0 < u < \pi:
Find the perpendicular height of the trapezium, h, in terms of u.
Find MP in terms of u.
Hence, form an equation for the area A of the trapezium MONP in terms of u.
Find the maximum possible area of the trapezium, correct to one decimal place.
We want to find the maximum perimeter P of the rectangle shown below. All measurements are in metres, and 0 \leq \theta \leq \dfrac{\pi}{2}:
Find an expression for x in terms of \theta.
Find an expression for y in terms of \theta.
Hence, write an expression for P, the perimeter of the rectangle, in terms of \theta.
Find the maximum perimeter of the rectangle correct to the nearest metre.
Consider the cone with a slant height of 3 units as shown in the diagram where 0 \lt \theta \lt \dfrac{\pi}{2}:
Write an expression for the radius of the base, r, in terms of \theta.
Write an expression for the perpendicular height, h, in terms of \theta.
Hence find an expression for the volume of the cone, V, in terms of \theta.
Find the size of angle \theta which maximises the volume. Round your answer correct to three decimal places.
Hence, find the maximum value of the volume of the cone, correct to three decimal places.
A ladder needs to be leaned against the side of a tall building. The ladder must go over the top of a tree that is 3 m high. The tree is located 6 m away from the building:
Find an expression for x in terms of \theta.
Find an expression for y in terms of \theta.
Hence, find an expression for the length of the ladder, L, in terms of \theta.
Find the minimum length of the ladder needed to go over the tree, correct to two decimal places.