The function y = a x^{2} - b x + c passes through the points (5, - 42) and (4, - 66) and has a maximum turning point at x = 3. Find the following:
\dfrac{dy}{dx}
a
c
b
The function f \left( x \right) = a x^{2} + \dfrac{b}{x^{2}} has turning points at x = 1 and x = - 1.
Use the fact that there is a turning point at x = 1 to form an equation for a in terms of b.
Use the fact that there is a turning point at x = - 1 to form an equation for a in terms of b.
What can you deduce about the values of a and b?
Consider the cubic function y = x^{3} - a x^{2} + b x + 11, which has stationary points at x=2 and x=10. Find the following:
\dfrac{dy}{dx}
a
b
The function f \left( x \right) = a x^{3} + b x^{2} + 9 x + 4 has a horizontal point of inflection at x = 1.
Write down the value of f'(1).
Hence, write an equation involving a and b.
Write down the value of f''(1).
Hence, write an equation for b in terms of a.
Find the value of a.
Find the value of b.
The function f \left( x \right) = a x^{3} + 9 x^{2} + c x + 11 has a turning point at x = 1 and a point of inflection at x = - 1.
Use the fact that there is a turning point at x = 1 to form an equation in terms of a and c.
Use the fact that there is a point of inflection at x = - 1 to solve for a.
Hence solve for the value of c.
The function f \left( x \right) = a x^{3} + 18 x^{2} + c x + 4 has turning points at x = 4 and x = 2. Find the value of:
a
c
Determine the values of the non-zero constants, a and b, for the following function, given it has a turning point at \left(0.25, 1\right):
f \left( x \right) = a x e^{ b x}The number of cars (N) parked in the drive through section of a fast food restaurant t hours after midnight is given by: N = \frac{18 t}{2 + t^{2}}
Find the number of cars parked in the drive through at 12:30 am.
Determine the rate of change of the number of cars parked in the drive through at time t.
Determine the average rate of change of the number of cars parked in the drive through in the second hour.
Find the exact time t when the number of cars in the drive through is decreasing most rapidly.
The world's largest swing is set up so that when you leave the platform, you swing through a valley. Your height h \left( x \right) above the ground, x \text { m} horizontally from the platform, is given by h \left( x \right) = 170 e^{\cos \left(\frac{\pi x}{220}\right)}where x is defined from 0 to the first time you return to the height of the platform.
Determine your exact height above the ground when you are standing on the platform.
Calculate the horizontal distance travelled, x, before you return to the height of the platform.
Find the x-values of the stationary points of h \left( x \right).
Find the value of x at which you are closest to the ground.
Hence find the maximum vertical distance you are located from the platform during a swing, correct to two decimal places.
A barrel is being filled and then emptied with water in such a way that the volume of water, V\text{ mL}, in the barrel after a time t seconds is given by V \left( t \right) = \frac{40}{3} t^{2} - \frac{1}{9} t^{3}for 0 \leq t \leq 120.
Find the exact average rate of flow in the first 20 seconds in millilitres per second.
Find a, the rate of flow at 20 seconds, which is in millilitres per second.
After reaching its maximum volume the barrel springs a large leak and empties. What was the exact maximum volume of the barrel, b in millilitres?
Determine when the rate of water flowing into the barrel is greatest.
Determine the exact rate of water flowing out of the barrel when the rate is fastest, in millilitres per second.
A population, P in thousands of insects, was observed to follow the model: P \left( t \right) = 1.5 \sin \left(\frac{\pi t}{6}\right) + 5 for 0 \leq t \leq 12 where t is the time in months after January 1st in a given year.
Find P' \left( t \right).
At what value of t is the population at its highest?
At what possible value(s) of t is the population increasing at the fastest rate?
What is the fastest rate of decrease in the population over the year? Give your answer in 1000 of insects per month.
What is the average rate of change in population between the population's peak and consecutive low? Give your answer in 1000 of insects per month.
The success of a product depends on the initial interest in the product in the first few weeks. A few marketing researchers use a special score to quantify the interest in a product, modelled by I \left( t \right) = 1590 t e^{ - 0.3 t } where t is the number of days since the product was available.
Find I' \left( t \right).
Hence find a, the maximum interest in the product.
What is the rate of change in interest after 10 days?
At what value of t does the rate of interest in the product start to decrease? Round your answer to two decimal places.
It is known that if a product has an interest score of at least 1000 for over 3 weeks, then the product will be mass produced. Will this product be mass produced?