Precalculus: Graphical, Numerical, Algebraic 8th Edition solutions

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find \(\Delta y / \Delta x\) on the interval \([1,4]\) under the given conditions. \(y=x^{2}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find \(\Delta y / \Delta x\) on the interval \([1,4]\) under the given conditions. \(y=\sqrt{x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find \(\Delta y / \Delta x\) on the interval \([1,4]\) under the given conditions. \(y=\log _{2} x\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find \(\Delta y / \Delta x\) on the interval \([1,4]\) under the given conditions. \(y=3^{x}\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find \(\Delta y / \Delta x\) on the interval \([1,4]\) under the given conditions. \(y=2 \text { when } x=1 \text { and } y=11 \text { when } x=4\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 16, find \(\Delta y / \Delta x\) on the interval \([1,4]\) under the given conditions. The graph of \(y=f(x)\) passes through points \((4,10)\) and \((1,-2)\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, compute the quotient \((f(1+h)-f(1-h)) / 2 h\) with the given and h. \(f(x)=\sin x, h=0.01\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, compute the quotient \((f(1+h)-f(1-h)) / 2 h\) with the given and h. \(f(x)=x^{4}, h=0.001\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, compute the quotient \((f(1+h)-f(1-h)) / 2 h\) with the given and h. \(f(x)=\ln x, h=0.001\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 710, compute the quotient \((f(1+h)-f(1-h)) / 2 h\) with the given and h. \(f(x)=e^{x}, h=0.0001\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=1-x^{2} \quad \text { at } \quad x=2\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=2 x+\frac{1}{2} x^{2} \quad \text { at } \quad x=2\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=3 x^{2}+2 \text { at } x=-2\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=x^{2}-3 x+1 \quad \text { at } \quad x=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=|x+2| \quad \text { at } \quad x=-2\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=\frac{1}{x+2} \quad \text { at } \quad x=-1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=\ln 2 x \quad \text { at } \quad x=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=2 \ln x \quad \text { at } \quad x=1\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=3 \sin x \quad \text { at } \quad x=\pi\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 110, use NDER on a calculator to find the numerical derivative of the function at the specified point. \(f(x)=\sin 3 x \quad \text { at } \quad x=\pi\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=x^{2},[0,4]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=x^{2},[-4,0]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=\sin x,[0, \pi]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=\sin x,[\pi, 2 \pi]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=\cos x,[0, \pi]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=|\cos x|,[0, \pi]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=1 / x,[1, e]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=1 / x,[e, 2 e]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=\frac{2}{1+x^{2}},\left[0,10^{8}\right]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 1120, use NINT on a calculator to find the numerical integral of the function over the specified interval. \(f(x)=\sec ^{2} x-\tan ^{2} x,[0,10]\)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Travel Time A truck is driven at a variable rate for 3 hours so that its velocity at any time \(t(0 \leq t \leq 3)\) is given by \(v(t)=35-12 \cos 4 t\) miles per hour. How far does the truck travel during the 3 hours? Round your answer to the nearest hundredth.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Travel Time A bicyclist rides for 90 minutes, and her velocity at any time t hours \((0 \leq t \leq 1.5)\) is given by \(v(t)=12-8 \sin 5 t\) miles per hour. How far does she travel during the 90 minutes? Round your answer to the nearest hundredth.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Finding Derivatives from Data A ball is dropped from the roof of a 30-story building. The height in feet above the ground of the falling ball is measured at 1/2-second intervals and recorded in the table on the next page. (a) Use the average velocity on the interval \([1,2]\) to estimate

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Estimating Average Rate of Change from Data Table 10.9 gives the U.S. gross domestic product data in billions of dollars for the years 19992007. (a) Find the average rate of change of the gross domestic product from 2001 to 2002 and then from 2005 to 2006. (b) Find a quadratic regres

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Estimating Velocity Refer to the data in Exercise 23. (a) Compute the average velocity, \(\Delta y / \Delta x\), on each subinterval of length 0.5. Make a table showing the midpoints of the subintervals in one column and the average velocities in the second column. (b) Make a scatter plot s

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Approximating Rate of Change Refer to the data in Exercise 24. (a) Compute the average rate of change, \(\Delta y / \Delta x\), on each subinterval. Make a table showing the midpoints of the subintervals in one column and the average rates of change in the second column. (b) Make a scatter

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Estimating Distance A stone is dropped from a cliff and its velocity (in feet per second) at regular 0.5-second intervals is shown in Table 10.10. Estimate the distance that the stone travels from \(t=0 \text { to } t=2.5\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Estimating Distance Table 10.11 shows the velocity of a moving object in meters per second, measured at regular 0.2- second intervals. Estimate the distance traveled by the body from \(t=0 \text { to } t=1.6\).

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Analyze the following program, which produces an LRAM approximation for the function entered in Y1 in the calculator. Then write a short paragraph explaining how it works.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Analyze the following program, which produces an RRAM approximation for the function entered in Y1 in the calculator. Then write a short paragraph explaining how it works.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 3142, complete the following for the indicated interval \([a, b]\). (a) Verify the given function is nonnegative. (b) Use a calculator to find the LRAM, RRAM, and average approximations for the area under the graph of the function from \(x=a \text { to } x=b\) with 10, 20, 50, a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The numerical derivative algorithm NDER always uses the same value of \(\Delta x(\text { or } h)\) to complete its calculations. Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
True or False The numerical integral algorithm always uses the same value of \(\Delta x\) to complete its calculations. Justify your answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4548, choose the correct answer. Do not use a calculator. Multiple Choice Estimating Area Under a Curve Which of the following will typically produce the most accurate estimate of an area under a curve? (A) NDER (B) NINT (C) LRAM, 10 rectangles (D) RRAM, 25 rectang

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4548, choose the correct answer. Do not use a calculator. Multiple Choice Estimating Derivative Values Given a continuous function , which of the following expressions will typically produce the most accurate estimate of \(f^{\prime}(a)\)? \(\begin{array}{l}\text{(A) } \frac{f(a

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4548, choose the correct answer. Do not use a calculator. Multiple Choice Using a Numerical Integral Which of the following cannot be estimated using a numerical integral? (A) The area under a curve that represents some function \(f(x)\) (B) The distance traveled, when the v

Precalculus: Graphical, Numerical, Algebraic 8th Edition
In Exercises 4548, choose the correct answer. Do not use a calculator. Multiple Choice Using a Numerical Derivative Which of the following cannot be estimated using a numerical derivative? (A) The instantaneous velocity of an object, when the position function is known (B) The slope of

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Let \(f(x)=2 x^{2}+3 x+1 \text { and } g(x)=x^{3}+1\). (a) Compute the derivative of . (b) Compute the derivative of g. (c) Using \(x=2 \text { and } h=0.001\), compute the standard difference quotient \(\frac{f(x+h)-f(x)}{h}\) and the symmetric difference quotient \(\frac{f(x+h)-f(x-h

Precalculus: Graphical, Numerical, Algebraic 8th Edition
When Are Derivatives and Areas Equal? Let \(f(x)=1+e^{x}\). (a) Draw a graph of f for \(0 \leq x \leq 1\). (b) Use NDER on your calculator to compute the derivative of at 1. (c) Use NINT on your calculator to compute the area under from \(x=0 \text { to } x=1\) to part (b). (d)

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Calculator Failure Many calculators report that NDER of \(f(x)=|x|\) evaluated at \(x=0\) is equal to 0. Explain why this is incorrect. Explain why this error occurs.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Grapher Failure Graph the function \(f(x)=|x| / x\) in the window \([-5,5] \text { by }[-3,3]\) and explain why \(f^{\prime}(0)\) does not exist. Find the value of NDER \(f(0)\) on the calculator and explain why it gives an incorrect answer.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Group Activity Finding Total Area The total area bounded by the graph of the function \(y=f(x)\) and the x-axis from \(x=a \text { to } x=b\) is the area below the graph of \(y=|f(x)| \text { from } x=a \text { to } x=b\) (a) Find the total area bounded by the graph of \(f(x)=\sin x\) and the x-

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn If a function is unbounded in an interval \([a, b]\) it may have finite area. Use your knowledge of limits at infinity to explain why this might be the case.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Writing to Learn Let and g be two continuous functions with \(f(x) \geq g(x)\) on an interval \([a, b]\). Devise a limit of sums definition of the area of the region between the two curves. Explain how to compute the area if the area under both curves is already known.

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Area as a Function Consider the function \(f(t)=2 t\). (a) Use NINT on a calculator to compute \(A(x)\) where A is the area under the graph of from \(t=0 \text { to } t=x \text { for } x=0.25\), 0.5, 1, 1.5, 2, 2.5, and 3. (b) Make a table of pairs \((x, A(x))\) for the values of x given i

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Area as a Function Consider the function \(f(t)=3 t^{2}\). (a) Use NINT on a calculator to compute \(A(x)\) where A is the area under the graph of from \(t=0 \text { to } t=x \text { for } x=0.25\), 0.5, 1, 1.5, 2, 2.5, and 3. (b) Make a table of pairs \((x, A(x))\) for the values of x giv

Precalculus: Graphical, Numerical, Algebraic 8th Edition
Group Activity Based on Exercises 56 and 57, discuss how derivatives (slope functions) and integrals (area functions) may be connected.