Concepts and Vocabulary
Multiple Choice If \(y=f( x) \) is a differentiable function, the differential \(dy=\) [(a) \(\Delta y,\) (b) \(\Delta x,\) (c) \(f(x) dx,\) (d) \(f^\prime(x) dx\)].
A linear approximation to a differentiable function \(f\) near \(x_{0}\) is given by the function \(L(x)=\) ______.
True or False The difference \(\vert \Delta y-dy\vert \) measures the departure of the graph of \(y=f( x) \) from the graph of the tangent line to \(f\).
If \(Q\) is a quantity to be measured and \(\Delta Q\) is the error made in measuring \(Q\), then the relative error in the measurement is given by the ratio ______.
True or False Newton’s Method uses tangent lines to the graph of \(f\) to approximate the zeros of \(f\).
True or False Before using Newton’s Method, we need a first approximation for the zero.
Skill Building
In Problems 7–16, find the differential \(dy\) of each function.
\(y=x^{3}-2x+1\)
\(y=e^{x}+2x-1\)
\(y=4(x^{2}+1)^{3/2}\)
\(y=\sqrt{x^{2}-1}\)
\(y=3\sin (2x) +x\)
\(y=\cos ^{2}(3x) -x\)
\(y=e^{-x}\)
\(y=e^{\sin x}\)
\(y=xe^{x}\)
\(y=\dfrac{e^{-x}}{x}\)
In Problems 17–22:
\(f(x) =e^{x}\) at \(x=1\)
\(f(x) =e^{-x}\) at \(x=1\)
\(f(x) =x^{2/3}\) at \(x=2\)
\(f(x) =x^{-1/2}\) at \(x=1\)
\(f(x) =\cos x\) at \(x=\pi \)
\(f(x) =\tan x\) at \(x=0\)
In Problems 23–30:
\(f(x)=(x+1) ^{5},\quad x_{0}=2\)
\(f(x)=x^{3}-1,\quad x_{0}=0\)
\(f(x)=\sqrt{x},\quad x_{0}=4\)
\(f(x)=x^{2/3},\quad x_{0}=1\)
\(f(x)=\ln x,\quad x_{0}=1\)
\(f(x) =e^{x},\) \(x_{0}=1\)
\(f(x)=\cos x,\quad x_{0}=\dfrac{\pi }{3}\)
\(f(x)=\sin x,\quad x_{0}=\dfrac{\pi }{6}\)
Approximate the change in:
Approximate the change in:
In Problems 33–40, for each function:
\(f(x) = x^{3}+3x-5\), interval: \((1, 2).\) Let \(c_{1}=1.5.\)
\(f(x) = x^{3}-4x+2\), interval: \((1,2).\) Let \(c_{1}=1.5\).
\(f(x) = 2x^{3}+3x^{2}+4x-1\), interval: \((0,1).\) Let \(c_{1}=0.5.\)
\(f(x) = x^{3}-x^{2}-2x+1\), interval: \((0,1) .\) Let \(c_{1}=0.5\)
\(f(x) = x^{3}-6x-12\), interval: \(( 3,4) .\) Let \(c_{1}=3.5.\)
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\(f(x) = 3x^{3}+5x-40\), interval: \(( 2,3) .\) Let \(c_{1}=2.5.\)
\(f(x) = x^{4}-2x^{3}+21x-23\), interval: \((1,2).\) Use a first approximation \(c_{1}\) of your choice.
\(f(x) = x^{4}-x^{3}+x-2\), interval: \((1,2).\) Use a first approximation \(c_{1}\) of your choice.
In Problems 41–46, for each function:
\(f(x) =x+e^{x},\) interval: \((-1,0) \)
\(f(x)=x-e^{-x},\) interval: \((0,1) \)
\(f( x) =x^{3}+\cos ^{2}x,\) interval: \((-1,0) \)
\(f( x) =x^{2}+2\sin x-0.5,\) interval: \((0,1) \)
\(f(x) =5-\sqrt{x^{2}+2},\) interval: \((4,5) \)
\(f( x) =2x^{2}+x^{2/3}-4,\) interval: \((1,2) \)
Applications and Extensions
Area of a Disk A circular plate is heated and expands. If the radius of the plate increases from \(R=10\) cm to \(R=10.1\) cm, use differentials to approximate the increase in the area of the top surface.
Volume of a Cylinder In a wooden block \(3\) cm thick, an existing circular hole with a radius of \(2\) cm is enlarged to a hole with a radius of \(2.2\) cm. Use differentials to approximate the volume of wood that is removed.
Volume of a Balloon Use differentials to approximate the change in volume of a spherical balloon of radius \(3\) m as the balloon swells to a radius of \(3.1\) m.
Volume of a Paper Cup A manufacturer produces paper cups in the shape of a right circular cone with a radius equal to one-fourth its height. Specifications call for the cups to have a top diameter of 4 cm. After production, it is discovered that the diameter measures only 3.8 cm. Use differentials to approximate the loss in capacity of the cup.
Volume of a Sphere
Distance Traveled A bee flies around a circle traced on an equator of a ball with a radius of 7 cm at a constant distance of 2 cm from the ball. An ant travels along the same circle but on the ball.
Estimating Height To find the height of a building, the length of the shadow of a 3-m pole placed 9 m from the building is measured. See the figure. This measurement is found to be 1 m, with a percentage error of 1%. Use differentials to approximate the height of the building. What is the percentage error in the estimate?
Pendulum Length The period of the pendulum of a grandfather clock is \(T=2\pi \sqrt{\dfrac{l}{g}}\), where \(l\) is the length (in meters) of the pendulum, \(T\) is the period (in seconds), and \(g\) is the acceleration due to gravity (\(9.8{\,{\rm{m}}}/\!{\,{\rm{s}}}^{2}\)). Suppose an increase in temperature increases the length \(l\) of the pendulum, a thin wire, by 1%. What is the corresponding percentage error in the period? How much time will the clock lose (or gain) each day?
Pendulum Length Refer to Problem 54. If the pendulum of a grandfather clock is normally 1 m long and the length is increased by \(10 \) cm, use differentials to approximate the number of minutes the clock will lose (or gain) each day.
Luminosity of the Sun The luminosity \(L\) of a star is the rate at which it radiates energy. This rate depends on the temperature\(\ T\) (in Kelvin, where \(0{\,{\rm{K}}}\) is absolute zero) and the surface area \(A\) of the star’s photosphere (the gaseous surface that emits the light). Luminosity at time \(t\) is given by the formula \(L(t)=\sigma AT^{4}\) , where \(\sigma \) is a constant, known as the Stefan–Boltzmann constant.
As with most stars, the Sun’s temperature has gradually increased over the 5 billion years of its existence, causing its luminosity to slowly increase. For this problem, we assume that increased luminosity \(L\) is due only to an increase in temperature \(T\). That is, we treat \(A\) as a constant.
Climbing a Mountain Weight \(W\) is the force on an object due to the pull of gravity. On Earth, this force is given by Newton’s Law of Universal Gravitation: \(W=\dfrac{GmM}{r^{2}}\), where \(m\) is the mass of the object, \(M=5.974\times 10^{24}{\,{\rm{kg}}}\) is the mass of Earth, \(r\) is the distance of the object from the center of the Earth, and \(G=6.67\times 10^{-11}{\,{\rm{m}}}^{3}/( {\,{\rm{kg}}}\cdot {\,{\rm{s}}}^{2}) \) is the universal gravitational constant. Suppose a person weighs \(70{\,{\rm{kg}}}\) at sea level, that is, when \(r=6370{\,{\rm{km}}}\) (the radius of Earth). Use differentials to
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approximate the person’s weight at the top of Mount Everest, which is \(8.8\) \({\,{\rm{km}}}\) above sea level.
Body Mass Index The body mass index (BMI) is given by the formula \[ {\rm BMI} = 703\dfrac{m}{h^{2}} \] where \(m\) is the person’s weight in pounds and \(h\) is the person’s height in inches. A BMI of \(25\) or less indicates that weight is normal, whereas a BMI greater than \(25\) indicates that a person is overweight.
Percentage Error The radius of a spherical ball is found by measuring the volume of the sphere (by finding how much water it displaces). It is determined that the volume is 40 cubic centimeters (cm\(^3\)), with a tolerance of 1%. Find the percentage error in the radius of the sphere caused by the error in measuring the volume.
Percentage Error The oil pan of a car has the shape of a hemisphere with a radius of \(8{\,{\rm{cm}}}\). The depth \(h\) of the oil is measured at \(3{\,{\rm{cm}}}\), with a percentage error of 10%. Approximate the percentage error in the volume. [Hint: The volume \(V\) for a spherical segment is \(V=\dfrac{1}{3}\pi h^{2}(3R-h)\), where \(R\) is the radius of the sphere.]
Percentage Error If the percentage error in measuring the edge of a cube is 2%, what is the percentage error in computing its volume?
Focal Length To photograph an object, a camera’s lens forms an image of the object on the camera’s photo sensors. A camera lens can be approximated by a thin lens, which obeys the thin-lens equation \(\dfrac{1}{f}=\dfrac{1}{p}+\dfrac{1}{q}\), where \(p\) is the distance from the lens to the object being photographed, \(q\) is the distance from the lens to the image of the object, and \(f\) is the focal length of the lens. A camera whose lens has a focal length of \(50{\,{\rm{mm}}}\) is being used to photograph a dog. The dog is originally \(15{\,{\rm{m}}}\) from the lens, but moves \(0.33{\,{\rm{m}}}\) (about a foot) closer to the lens. Use differentials to approximate the distance the image of the dog moved.
Using Newton’s Method to Solve Equations In Problems 63–66, use Newton’s Method to solve each equation correct to three decimal places.
\(e^{-x}=\ln x\)
\(e^{-x}=x-4\)
\(e^{x}=x^{2}\)
\(e^{x}=2\cos x\), \(x > 0\)
Approximating \(e\) Use Newton’s Method to approximate the value of \(e\) by finding the zero of the equation \(\ln x-1=0.\) Use \(c_{1}=3\) as the first approximation and find the fourth approximation to the zero. Compare the results from this approximation to the value of \(e\) obtained with a calculator.
Show that the linear approximation of a function \(f( x) =( 1+x) ^{k}\), where \(x\) is near \(0\) and \(k\) is any number, is given by \(y=1+kx\).
Does it seem reasonable that if a first degree polynomial approximates a differentiable function in an interval near \(x_{0}\), a higher-degree polynomial should approximate the function over a wider interval? Explain your reasoning.
Why does a function need to be differentiable at \(x_{0}\) for a linear approximation to be used?
Newton’s Method Suppose you use Newton’s Method to solve \(f( x) =0\) for a differentiable function \(f,\) and you obtain \(x_{n+1}=x_{n}.\) What can you conclude?
When Newton’s Method Fails Verify that the function \(f(x)= -x^{3}+ 6x^{2}-9x+6\) has a zero in the interval \((2,5)\). Show that Newton’s Method fails if an initial estimate of \(c_{1}=2.9\) is chosen. Repeat Newton’s Method with an initial estimate of \(c_{1}=3.0\). Explain what occurs for each of these two choices. (The zero is near \(x=4.2\).)
When Newton’s Method Fails Show that Newton’s Method fails if it is applied to \(f(x)=x^{3}-2x+2\) with an initial estimate of \(c_{1}=0\).
When Newton’s Method Fails Show that Newton’s Method fails if \(f(x)=x^{8}-1\) if an initial estimate of \(c_{1}=0.1\) is chosen. Explain what occurs.
When Newton’s Method Fails Show that Newton’s Method fails if it is applied to \(f(x)=(x-1)^{1/3}\) with an initial estimate of \(c_{1}=2\).
Newton’s Method
Challenge Problems
Specific Gravity A solid wooden sphere of diameter \(d\) and specific gravity \(S\) sinks in water to a depth \(h\), which is determined by the equation \(2x^{3}-3x^{2}-S=0\), where \(x=\dfrac{h}{d}\). Use Newton’s Method to find a third approximation to \(h\) for a maple ball of diameter 6 in. for which \(S=0.786\).
Kepler’s Equation The equation \(x-p\sin x=M\), called Kepler’s equation, occurs in astronomy. Use Newton’s Method to find a second approximation to \(x\) when \(p=0.2\) and \(M=0.85\). Use \(c_{1}=1\) as your first approximation.