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.
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 above the ground, H 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 track 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 track 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 to follow the model: P(t)= 2.5 \sin (\dfrac{\pi t}{6}) +4 for 0 \leq t \leq 12 over the course of a year, with time, t, 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?
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 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 exact value of d that will give a beam of maximum strength.
Hence, find the maximum strength of a rectangular beam in terms of k.
A square beam with diagonal \sqrt{27} \text{ cm} is to be cut from an identical log. Find the exact value of 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 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.
If the side lengths of the base measure x \text { cm}, and the height of the box measures h \text { cm}, express h in terms of x.
Let the surface area of the open box be represented by S. Find an equation for S.
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 (c).
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 the nearest whole \text{m}^{3}.
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 numbers of the months in which the maximum price is achieved, and the minimum price is achieved.
Calculate the overall change in the share price between the minimum and maximum share price during the 12 months, correct to two decimal places.
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.
Find the possible exact values of x that could maximise the area of the rectangle.
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.
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 the points 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} |
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.
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 a CAS calculator, 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.
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.
By differentiating A with respect to \theta, and then using a CAS calculator, find the possible values of \theta (in radians) which could minimise the area. Round your answer correct to three decimal places.
What can we conclude about A when \theta = answer to part (d)?
Hence, find the minimum area of the triangle she has to cut off the scarf to remove the burn. Round your answer rounded 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.
Hence, write an expression for a in terms of \theta.
Find an expression for the total length across the corridor, L, in terms of \theta.
By differentiating L with respect to \theta, and then using a CAS calculator, find the values of \theta for which the length across the corridor could be at its minimum. Round your answer correct to three decimal places.
What can we conclude about L when \theta = 0.823?
Hence, Find 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 the 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.
By differentiating A with respect to \theta, and then using a CAS calculator, find the possible values of \theta which could maximise the area of the cross-section of the trough and hence maximise the amount of water it can hold. Give your answer as an exact value.
What can we conclude about A when \theta = \dfrac{\pi}{3}?
Hence, 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.
Differentiate T with respect to \theta and then use a CAS calculator to solve for the value of \theta which could maximise or minimise the time taken on this route. Round your answer correct to three decimal places.
What can we conclude about T when \theta = answer to part (d)?
Hence, find the maximum time Georgia can take on the route, correct to three decimal places.
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.
By differentiating P with respect to \theta, and then using a CAS calculator to solve, find the value of \theta which maximises the perimeter of the rectangle. Round your answer to three decimal places.
What can we conclude about A when \theta = 0.464?
Hence, 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 long 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.
By differentiating L with respect to \theta, and then using a CAS calculator, find the value of \theta which could minimise the length. Round your answer to three decimal places.
What can we conclude about L when \theta = 0.671?
Hence, find the minimum length of the ladder needed to go over the tree. Round your answer to two decimal places.