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The esteemed author team is back with a fourth edition of Calculus: Graphing, Numerical, Algebraic written specifically for high school students and aligned to the guidelines of the AP® Calculus exam. The new edition focuses on providing enhanced student and teacher support; for students, the authors added guidance on the appropriate use of graphing calculators and updated exercises to reflect current data. For teachers, the authors provide lesson plans, pacing guides, and point-of-need answers throughout the Teacher's Edition and teaching resources.
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Sample questions asked in the 4th edition of Calculus: Graphical, Numerical, Algebraic:
Volume of a Bowl A bowl has a shape that can be generated by revolving the graph of y = x 2 /2 between y = 0 and y = 5 about the y -axis. (a) Find the volume of the bowl. (b) If we fill the bowl with water at a constant rate of 3 cubic units per second, how fast will the water level in the bowl be rising when the water is 4 units deep?
Tolerance (a) About how accurately must the interior diameter of a 10-m-high cylindrical storage tank be measured to calculate the tank’s volume to within 1% of its true value? (b) About how accurately must the tank’s exterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5% of the true amount?
(a) Sketch a graph of the solution to the initial value problem ? (b) Writing to Learn A student solved part (a) and used a graphing calculator to produce the following graph: How would you explain to this student why this graph is not the correct answer to part (a)?
Group Activity Inverse Functions Let y = f ( x ) = mx + b , m ? 0. (a) Writing to Learn Give a convincing argument that f is a one-to-one function. (b) Find a formula for the inverse of f . How are the slopes of f and f ?1 related? (c) If the graphs of two functions are parallel lines with a nonzero slope, what can you say about the graphs of the inverses of the functions? (d) If the graphs of two functions are perpendicular lines with a nonzero slope, what can you say about the graphs of the inverses of the functions?
Use Equation 1 from Exercise 42. Tennis During the match in which Pete Sampras won the 1990 U.S. Open men’s tennis championship, Sampras hit a serve that was clocked at a phenomenal 124 mph. How much work did Sampras have to do on the 2-oz ball to get it to that speed?
Oil Flow Oil flows through a cylindrical pipe of radius 3 inches, but friction from the pipe slows the flow toward the outer edge. The speed at which the oil flows at a distance r inches from the center is 8(10 ? r 2 ) inches per second. (a) In a plane cross section of the pipe, a thin ring with thickness ? r at a distance r inches from the center approximates a rectangular strip when you straighten it out. What is the area of the strip (and hence the approximate area of the ring)? (b) Explain why we know that oil passes through this ring at approximately 8(10 ? r 2 )(2 ?r ) ? r cubic inches per second. (c) Set up and evaluate a definite integral that will give the rate (in cubic inches per second) at which oil is flowing through the pipe.
Learn more.
Sample questions asked in the 4th edition of Calculus: Graphical, Numerical, Algebraic:
Volume of a Bowl A bowl has a shape that can be generated by revolving the graph of y = x 2 /2 between y = 0 and y = 5 about the y -axis. (a) Find the volume of the bowl. (b) If we fill the bowl with water at a constant rate of 3 cubic units per second, how fast will the water level in the bowl be rising when the water is 4 units deep?
Tolerance (a) About how accurately must the interior diameter of a 10-m-high cylindrical storage tank be measured to calculate the tank’s volume to within 1% of its true value? (b) About how accurately must the tank’s exterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5% of the true amount?
(a) Sketch a graph of the solution to the initial value problem ? (b) Writing to Learn A student solved part (a) and used a graphing calculator to produce the following graph: How would you explain to this student why this graph is not the correct answer to part (a)?
Group Activity Inverse Functions Let y = f ( x ) = mx + b , m ? 0. (a) Writing to Learn Give a convincing argument that f is a one-to-one function. (b) Find a formula for the inverse of f . How are the slopes of f and f ?1 related? (c) If the graphs of two functions are parallel lines with a nonzero slope, what can you say about the graphs of the inverses of the functions? (d) If the graphs of two functions are perpendicular lines with a nonzero slope, what can you say about the graphs of the inverses of the functions?
Use Equation 1 from Exercise 42. Tennis During the match in which Pete Sampras won the 1990 U.S. Open men’s tennis championship, Sampras hit a serve that was clocked at a phenomenal 124 mph. How much work did Sampras have to do on the 2-oz ball to get it to that speed?
Oil Flow Oil flows through a cylindrical pipe of radius 3 inches, but friction from the pipe slows the flow toward the outer edge. The speed at which the oil flows at a distance r inches from the center is 8(10 ? r 2 ) inches per second. (a) In a plane cross section of the pipe, a thin ring with thickness ? r at a distance r inches from the center approximates a rectangular strip when you straighten it out. What is the area of the strip (and hence the approximate area of the ring)? (b) Explain why we know that oil passes through this ring at approximately 8(10 ? r 2 )(2 ?r ) ? r cubic inches per second. (c) Set up and evaluate a definite integral that will give the rate (in cubic inches per second) at which oil is flowing through the pipe.
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