The measurement of the side of a square floor tile is 10 inches, with a possible error

In Exercises 1-6, find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. \(f(x)=x^{2}, \quad (2, 4)\) Text Transcription: f(x)=x^2, (2, 4)

In Exercises 1-6, find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. \(f(x)=\frac{6}{x^{2}}, \quad\left(2, \frac{3}{2}\right)\) Text Transcription: f(x)=6/x^2, (2, 3/2)

In Exercises 1-6, find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. \(f(x)=x^{5}, \quad (2, 32)\) Text Transcription: f(x)=x^5, (2, 32)

In Exercises 1-6, find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. \(f(x)=\sqrt{x}, \quad(2, \sqrt{2})\) Text Transcription: f(x)=sqrt x, (2, sqrt 2)

In Exercises 1-6, find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. \(f(x)=\sin x, \quad(2, \sin 2)\) Text Transcription: f(x)=sin x,(2, sin 2)

In Exercises 1-6, find the equation of the tangent line T to the graph of f at the given point. Use this linear approximation to complete the table. \(f(x)=\csc x, \quad(2, \csc 2)\) Text Transcription: f(x)=csc x,(2, csc 2)

In Exercises 7-10, use the information to evaluate and compare \(\Delta y\) and dy. \(y=x^{3} \quad x=1 \quad \Delta x=d x=0.1\) Text Transcription: Delta y y=x^3 x=1 Delta x=d x=0.1

In Exercises 7-10, use the information to evaluate and compare \(\Delta y\) and dy. \(y=1-2 x^{2} \quad x=0 \quad \Delta x=d x=-0.1\) Text Transcription: Delta y y=1-2 x^2 x=0 Delta x=d x=-0.1

In Exercises 7-10, use the information to evaluate and compare \(\Delta y\) and dy. \(y=x^{4}+1 \quad x=-1 \quad \Delta x=d x=0.01\) Text Transcription: Delta y y=x^4+1 x=-1 Delta x=d x=0.01

In Exercises 7-10, use the information to evaluate and compare \(\Delta y\) and dy. \(y=2-x^{4} \quad x=2 \quad \Delta x=d x=0.01\) Text Transcription: Delta y y=2-x^4 x=2 Delta x=d x=0.01

In Exercises 11–20, find the differential dy of the given function. \(y=3 x^{2}-4\) Text Transcription: y=3 x^2-4

In Exercises 11–20, find the differential dy of the given function. \(y=3 x^{2 / 3}\) Text Transcription: y=3 x^2/3

In Exercises 11–20, find the differential dy of the given function. \(y=\frac{x+1}{2 x-1}\) Text Transcription: y=x+1/2x+1

In Exercises 11–20, find the differential dy of the given function. \(y=\sqrt{9-x^{2}}\) Text Transcription: y=sqrt 9-x^2

In Exercises 11–20, find the differential dy of the given function. \(y=x \sqrt{1-x^{2}}\) Text Transcription: y=x sqrt 1-x^2

In Exercises 11–20, find the differential dy of the given function. \(y=\sqrt{x}+\frac{1}{\sqrt{x}}\) Text Transcription: y=sqrt x + 1/sqrt x

In Exercises 11–20, find the differential dy of the given function. \(y=3 x-\sin ^{2} x\) Text Transcription: y=3 x-sin^2 x

In Exercises 11–20, find the differential dy of the given function. \(y=x \cos x\) Text Transcription: y=x cos x

In Exercises 11–20, find the differential dy of the given function. \(y=\frac{1}{3} \cos \left(\frac{6 \pi x-1}{2}\right)\) Text Transcription: y=1/3 cos (6 pi x-1/2)

In Exercises 11–20, find the differential dy of the given function. \(y=\frac{\sec ^{2} x}{x^{2}+1}\) Text Transcription: y=sec^2 x/x^2+1

In Exercises 21-24, use differentials and the graph of f to approximate (a) f(1.9) and (b) f(2.04). To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 21-24, use differentials and the graph of f to approximate (a) f(1.9) and (b) f(2.04). To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 21-24, use differentials and the graph of f to approximate (a) f(1.9) and (b) f(2.04). To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 21-24, use differentials and the graph of f to approximate (a) f(1.9) and (b) f(2.04). To print an enlarged copy of the graph, go to the website www.mathgraphs.com.

In Exercises 25 and 26, use differentials and the graph of g' to approximate (a) g(2.93) and (b) g(3.1) given that g(3) = 8.

In Exercises 25 and 26, use differentials and the graph of g' to approximate (a) g(2.93) and (b) g(3.1) given that g(3) = 8.

Area The measurement of the side of a square floor tile is 10 inches, with a possible error of \(\frac{1}{32}\) inch. Use differentials to approximate the possible propagated error in computing the area of the square. Text Transcription: 1/32

Area The measurements of the base and altitude of a triangle are found to be 36 and 50 centimeters, respectively. The possible error in each measurement is 0.25 centimeter. Use differentials to approximate the possible propagated error in computing the area of the triangle.

Area The measurement of the radius of the end of a log is found to be 16 inches, with a possible error of \(\frac{1}{4}\) inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log. Text Transcription: 1/4

Volume and Surface Area The measurement of the edge of a cube is found to be 15 inches, with a possible error of 0.03 inch. Use differentials to approximate the maximum possible propagated error in computing (a) the volume of the cube and (b) the surface area of the cube.

Area The measurement of a side of a square is found to be 12 centimeters, with a possible error of 0.05 centimeter. (a) Approximate the percent error in computing the area of the square. (b) Estimate the maximum allowable percent error in measuring the side if the error in computing the area cannot exceed 2.5%.

Circumference The measurement of the circumference of a circle is found to be 64 centimeters, with a possible error of 0.9 centimeter. (a) Approximate the percent error in computing the area of the circle. (b) Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed 3%.

Volume and Surface Area The radius of a spherical balloon is measured as 8 inches, with a possible error of 0.02 inch. Use differentials to approximate the maximum possible error in calculating (a) the volume of the sphere, (b) the surface area of the sphere. and (c) the relative errors in parts (a) and (b).

Stopping Distance The total stopping distance T of a vehicle is \(T=2.5 x+0.5 x^{2}\) where T is in feet and x is the speed in miles per hour. Approximate the change and percent change in total stopping distance as speed changes from x = 25 to x = 26 miles per hour. Text Transcription: T=2.5x+0.5x^2

Volume In Exercises 35 and 36, the thickness of each shell is 0.2 centimeter. Use differentials to approximate the volume of each shell.

Volume In Exercises 35 and 36, the thickness of each shell is 0.2 centimeter. Use differentials to approximate the volume of each shell.

Pendulum The period of a pendulum is given by \(T=2 \pi \sqrt{\frac{L}{g}}\) where L is the length of the pendulum in feet, g is the acceleration due to gravity, and T is the time in seconds. The pendulum has been subjected to an increase in temperature such that the length has increased by \(\frac{1}{2} \%\). (a) Find the approximate percent change in the period. (b) Using the result in part (a), find the approximate error in this pendulum clock in 1 day. Text Transcription: T=2pi sqrt L/g 1/2%

Ohm's Law A current of I amperes passes through a resistor of R ohms. Ohm's Law states that the voltage E applied to the resistor is E = IR. If the voltage is constant, show that the magnitude of the relative error in R caused by a change in I is equal in magnitude to the relative error in I.

Triangle Measurements The measurement of one side of a right triangle is found to be 9.5 inches, and the angle opposite that side is \(26^{\circ} 45^{\prime}\) with a possible error of 15'. (a) Approximate the percent error in computing the length of the hypotenuse. (b) Estimate the maximum allowable percent error in measuring the angle if the error in computing the length of the hypotenuse cannot exceed 2%. Text Transcription: 26 degree 45’

Area Approximate the percent error in computing the area of the triangle in Exercise 39.

Projectile Motion The range R of a projectile is \(R=\frac{v_{0}^{2}}{32}(\sin 2 \theta)\) where \(v_{0}\) is the initial velocity in feet per second and \(\theta\) is the angle of elevation. If \(v_{0}=2500\) feet per second and \(\theta\) is changed from \(10^{\circ} \text { to } 11^{\circ}\), use differentials to approximate the change in the range. Text Transcription: R=v_0^2/32(sin 2 theta) v_0 theta 10 degrees to 11 degrees v_0=2500

Surveying A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as \(71.5^{\circ}\). How accurately must the angle be measured if the percent error in estimating the height of the tree is to be less| than 6%? Text Transcription: 71.5 degrees

In Exercises 43-46, use differentials to approximate the value of the expression. Compare your answer with that of a calculator. \(\sqrt{99.4}\) Text Transcription: sqrt 99.4

In Exercises 43-46, use differentials to approximate the value of the expression. Compare your answer with that of a calculator. \(\sqrt[3]{26}\) Text Transcription: 3 sqrt 26

In Exercises 43-46, use differentials to approximate the value of the expression. Compare your answer with that of a calculator. \(\sqrt[4]{624}\) Text Transcription: 4 sqrt 624

In Exercises 43-46, use differentials to approximate the value of the expression. Compare your answer with that of a calculator. \((2.99)^{3}\) Text Transcription: (2.99)^3

In Exercises 47 and 48, verify the tangent line approximation of the function at the given point. Then use a graphing utility to graph the function and its approximation in the same viewing window. Function Approximation Point \(f(x)=\sqrt{x+4}\) \(y=2+\frac{x}{4}\) (0, 2) Text Transcription: f(x)=sqrt x+4 y=2+x/4

In Exercises 47 and 48, verify the tangent line approximation of the function at the given point. Then use a graphing utility to graph the function and its approximation in the same viewing window. Function Approximation Point f(x) = tan x y = x (0, 0)

Describe the change in accuracy of dy as an approximation for \(\Delta y\) when \(\Delta x\) is decreased. Text Transcription: Delta y Delta x

When using differentials, what is meant by the terms propagated error, relative error, and percent error?

Give a short explanation of why the approximation is valid. (a) \(\sqrt{4.02} \approx 2+\frac{1}{4}(0.02)\) (b) \(\tan 0.05 \approx 0+1(0.05)\) Text Transcription: sqrt 4.02 approx 2 + 1/4 (0.02) tan 0.05 approx 0 + 1(0.05)

Would you use y = x to approximate f(x) = sin x near x = 0? Why or why not?

True or False? In Exercises 53-56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y = x + c, then dy = dx.

True or False? In Exercises 53-56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If y = ax + b, then \(\Delta y / \Delta x=d y / d x\). Text Transcription: Delta y/Delta x=dy/dx

True or False? In Exercises 53-56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. if y is differentiable, then \(\lim _{\Delta x \rightarrow 0}(\Delta y-d y)=0\). Text Transcription: lim_Delta x rightarrow 0 (Delta y - dy)=0

True or False? In Exercises 53-56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. if y = f(x), f is increasing and differentiable, and \(\Delta x > 0\), then \(\Delta y \geq d y\)/ Text Transcription: Delta x > 0 Delta y geq dy