The cost C, in dollars, of producing x items of a product, is modelled by the function C \left( x \right) = 1300 + 7 x + 0.002 x^{2}
Determine the marginal cost function.
Hence calculate the marginal cost, C' \left( 90 \right), when 90 items have already been produced.
The profit P, in dollars, from producing and selling x items is modelled by the function P \left( x \right) = 2.3 x - 0.003 x^{2}
Determine the marginal profit function.
Hence calculate the marginal profit, P' \left( 100 \right), when 100 items have been sold.
The revenue R, in dollars, earned from selling x items is modelled by the function R \left( x \right) = - 0.02 x^{2} + 40 x + 4000
Determine the marginal revenue function.
Hence calculate the marginal revenue, R' \left( 80 \right), when 80 items have been sold.
Find the maximum number of items, x, that can be sold before revenue begins to fall.
The cost C in dollars, of producing x diodes is modelled by the function C \left( x \right) = 12 x + \dfrac{18\,050}{x}
Determine the marginal cost function.
Hence calculate the cost of producing the 96th diode.
The cost of producing x thousand books is given by \left(x + 1\right)^{2} - 1, measured in dollars. Each book is sold for \$22.79, so that x thousand books are sold for \$22\,790 x
Let P \left( x \right) be the profit from selling x thousand books. Determine the expression for marginal profit.
After how many thousand books is it no longer economically feasible to continue production?
Consider the function y = f \left( x \right) = - 0.6 x^{2} + 11.
Determine f' \left( x \right).
Use the increments formula to calculate the approximate change in y when x changes from 2 to 2.01.
Consider the function y = f \left( x \right) = \dfrac{4}{x^{2}} + \sqrt{x}.
Determine f' \left( x \right).
Use the increments formula to calculate the approximate change in y when x changes from 1 to 1.001. Express your answer as an exact value.
Consider the function y = f \left( x \right) = 6 x \left(x - 4\right)^{2}.
Determine f' \left( x \right).
Use the increments formula to calculate the approximate change in y when x changes from 2 to 2.02.
The cost, C dollars, of producing x items of a product, is modelled by the function C \left( x \right) = \sqrt{800 + \left(x + 10\right)^{2}}
Determine the marginal cost function.
Hence calculate the cost of producing the 61st item. Round your answer to two decimal places.
Consider the volume V \left( h \right) of a cone that has a radius measuring 7 \text{ cm} and a variable height h \text{ cm}.
Determine V' \left( h \right).
Use the increments formula to form an expression for the percentage change in the volume of a cone that corresponds to a 4\% increase in its height.
Consider the volume V of a balloon that is in the shape of a sphere of radius r \text{ cm}.
Determine V' \left( r \right).
Use the increments formula to find the approximate change in the volume of a spherical balloon when the radius changes from 4 \text{ cm} to 3.98 \text{ cm}. Round your answer to two decimal places.
Consider the surface area, S, of a cube that has side lengths measuring r \text{ cm}.
Determine S' \left( r \right).
Use the increments formula to find the approximate change in the surface area of a cube when the side length changes from 11 \text{ cm} to 11.01 \text{ cm}. Give your answer as an exact value.
The height of a certain tree can be modelled by H \left( t \right) = 18 - \dfrac{87}{2 t + 5}, where t is the time in years after the tree was planted from a nursery seedling and H is the height of the tree in metres.
Determine H' \left( t \right).
Use the increments formula to calculate the approximate change in height in metres when t changes from two to 2.25. Round your answer to two decimal places.
Hence, estimate the percentage change in height when t changes from 2 to 2.25. Round your answer to the nearest percent.
Approximate the change in time required for a growth of 10\text{ cm} when the tree is 1 year old. Round your answer to two decimal places.
Consider the function y = f \left( x \right) = 15 e^{ 2 x}.
Determine f' \left( x \right).
Use the increments formula to calculate the approximate change in y when x changes from 1 to 1.25.
Hence, estimate the percentage change in y when x changes from 1 to 1.25.
The decay of a radioactive substance can be modelled by W \left( t \right) = 250 e^{ - 0.05 t }, where t is the time in years and W is the weight of the substance in grams.
Determine W' \left( t \right).
Use the increments formula to calculate the approximate change in weight when t changes from 10 to 10.5.
Hence, estimate the percentage change in weight when t changes from 10 to 10.5.
The temperature of a cup of coffee placed on a bench to cool can be modelled by T \left( x \right) = 22 + 68 e^{ - 0.07 x }where x is the time in minutes and T is the temperature in degrees Celsius.
Determine T' \left( x \right).
Use the increments formula to calculate the approximate change in temperature when x changes from 5 to 5.4. Round your answer to two decimal places.
What is the actual change in temperature when x changes from 5 to 5.4? Round your answer to two decimal places.
What is the percentage error of your estimate given in part (b)? Round your percentage to two decimal places.
Consider the function y = f \left( x \right) = x e^{ 5 x}.
Determine f' \left( x \right).
Use the increments formula to calculate the approximate change in y when x changes from 1 to 1.2.
Hence, estimate the percentage change in y when x changes from 1 to 1.2.
Consider the function y = f \left( x \right) = \cos \left( 2 x\right).
Determine f' \left( x \right).
Use the increments formula to calculate the approximate change in y when x changes from 0 to 0.05.
Hence, estimate the percentage change in y when x changes from 0 to 0.05.
Consider the function f \left( x \right) = \sin \left(\dfrac{x}{2}\right).
Determine f' \left( x \right).
Use the increments formula to calculate the approximate change in y when x changes from 0 to 0.4.
What is the actual change in y when x changes from 0 to 0.4? Round your answer to four decimal places.
What is the percentage error of your estimate? Round your percentage to two decimal places.
In a particular bay, the height in metres of the tide above the mean sea level is given by H \left( t \right) = 4 \sin \left(\dfrac{\pi \left(t - 2\right)}{6}\right)where t is the time in hours since midnight.
Determine H' \left( t \right).
Use the increments formula to calculate the approximate change in height when t changes from 4 to 4.2.
Hence, estimate the percentage change in height when t changes from 4 to 4.2. Round your percentage to two decimal places.
Consider the function y = f \left( x \right) = \dfrac{6 x}{\left(10 - x\right)^{4}}.
Determine f' \left( x \right).
Use the increments formula to form an expression for the percentage change in y when x increases by 2\%.
Consider the surface area S of a spherical balloon that has a radius measuring r cm.
Determine S' \left( r \right).
Use the increments formula to form an expression for the percentage error in the surface area of a sphere that corresponds to an error of 2\% in the radius.
The period of oscillation T of a pendulum of length L is given by T = 2 \pi \sqrt{\dfrac{L}{g}}, where \pi and g are constants.
Determine T' \left( L \right).
Use the increments formula to find the approximate percentage change in T corresponding to a 5\% drop in the length of the pendulum.
Use the increments formula to find the approximate percentage change in L corresponding to a 3\% increase in the period of oscillation.
The capacity, C \text{ cm}^3, of a hemispherical bowl is given by C = \dfrac{2}{3}\pi r^3.
Find the radius of a bowl with a capacity of 300 \text{ cm}^3. Round your answer to two decimal places.
Use the increments formula to estimate the change in radius required for the capacity to increase from 300 \text{ cm}^3 to 310 \text{ cm}^3. Round your answer to three decimal places.
The volume, V \text{ cm}^3, of water poured into a spherical vessel of radius 30 \text{ cm} to a depth x is given by V = \dfrac{\pi x^2}{3}(90-x).
Use the increments formula to estimate the change in depth required for the volume to change from 20 \text{ cm}^3 to 60 \text{ cm}^3. Round your answer to three decimal places.