11.20 Small increments and approximations
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.