For positive numbers the pulse function is defined as

In Exercises 1 and 2, determine whether the problem can be solved using precalculus or if calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning. Use a graphical or numerical approach to estimate the solution. Find the distance between the points and along the curve 4x

In Exercises 1 and 2, determine whether the problem can be solved using precalculus or if calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning. Use a graphical or numerical approach to estimate the solution. Find the distance between the points and along the line 4x2

In Exercises 3 and 4, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. 4x2

In Exercises 3 and 4, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. x20

In Exercises 5 and 6, use the graph to determine each limit. x20y

In Exercises 5 and 6, use the graph to determine each limit. 20y

In Exercises 710, find the limit Then use the - definition to prove that the limit is L. x20y

In Exercises 710, find the limit Then use the - definition to prove that the limit is L. 20y

In Exercises 710, find the limit Then use the - definition to prove that the limit is L. 0y4

In Exercises 710, find the limit Then use the - definition to prove that the limit is L. 0y4

In Exercises 1124, find the limit (if it exists). y4

In Exercises 1124, find the limit (if it exists). 4x

In Exercises 1124, find the limit (if it exists). 4x

In Exercises 1124, find the limit (if it exists). x3

In Exercises 1124, find the limit (if it exists). x3xy

In Exercises 1124, find the limit (if it exists). x3xy2

In Exercises 1124, find the limit (if it exists). 3xy2

In Exercises 1124, find the limit (if it exists). 3xy210

In Exercises 1124, find the limit (if it exists). xy210xy

In Exercises 1124, find the limit (if it exists). 210xy

In Exercises 1124, find the limit (if it exists). 10xy4

In Exercises 1124, find the limit (if it exists). 10xy4y

In Exercises 1124, find the limit (if it exists). xy4y

In Exercises 1124, find the limit (if it exists). 4yx

In Exercises 25 and 26, evaluate the limit given and 4yx3

In Exercises 25 and 26, evaluate the limit given and yx3

Numerical, Graphical, and Analytic Analysis In Exercises 27 and 28, consider (a) Complete the table to estimate the limit. (b) Use a graphing utility to graph the function and use the graph to estimate the limit. (c) Rationalize the numerator to find the exact value of the limit analytically. x3x

Numerical, Graphical, and Analytic Analysis In Exercises 27 and 28, consider (a) Complete the table to estimate the limit. (b) Use a graphing utility to graph the function and use the graph to estimate the limit. (c) Rationalize the numerator to find the exact value of the limit analytically. x3xy

Free-Falling Object In Exercises 29 and 30, use the position function which gives the height (in meters) of an object that has fallen from a height of 200 meters. The velocity at time seconds is given by Find the velocity of the object when 3xy2

Free-Falling Object In Exercises 29 and 30, use the position function which gives the height (in meters) of an object that has fallen from a height of 200 meters. The velocity at time seconds is given by At what velocity will the object impact the ground? xy2

In Exercises 3136, find the limit (if it exists). If the limit does not exist, explain why xy2x

In Exercises 3136, find the limit (if it exists). If the limit does not exist, explain why y2x3

In Exercises 3136, find the limit (if it exists). If the limit does not exist, explain why 2x3

In Exercises 3136, find the limit (if it exists). If the limit does not exist, explain why x34

In Exercises 3136, find the limit (if it exists). If the limit does not exist, explain why x34x

In Exercises 3136, find the limit (if it exists). If the limit does not exist, explain why 34xy

In Exercises 3746, determine the intervals on which the function is continuous. 4xy

In Exercises 3746, determine the intervals on which the function is continuous. 4xyx

In Exercises 3746, determine the intervals on which the function is continuous. xyx2

In Exercises 3746, determine the intervals on which the function is continuous. yx2

In Exercises 3746, determine the intervals on which the function is continuous. x2x

In Exercises 3746, determine the intervals on which the function is continuous. x2xy

In Exercises 3746, determine the intervals on which the function is continuous. 2xy

In Exercises 3746, determine the intervals on which the function is continuous. xyx

In Exercises 3746, determine the intervals on which the function is continuous. xyx2

In Exercises 3746, determine the intervals on which the function is continuous. yx2

Determine the value of such that the function is continuous on the entire real line. x22

Determine the values of and such that the function is continuous on the entire real line. x22y

Use the Intermediate Value Theorem to show that has a zero in the interval 22y

Delivery Charges The cost of sending an overnight package from New York to Atlanta is $9.80 for the first pound and $2.50 for each additional pound or fraction thereof. Use the greatest integer function to create a model for the cost of overnight delivery of a package weighing pounds. Use a graphing utility to graph the function and discuss its continuity. 2y2

Let Find each limit (if possible). 2y2x

Let (a) Find the domain of (b) Find (c) Find y2x

In Exercises 5356, find the vertical asymptotes (if any) of the graphs of the function. 2x

In Exercises 5356, find the vertical asymptotes (if any) of the graphs of the function. 2xx

In Exercises 5356, find the vertical asymptotes (if any) of the graphs of the function. xx2

In Exercises 5356, find the vertical asymptotes (if any) of the graphs of the function. x2

In Exercises 5768, find the one-sided limit. x21

In Exercises 5768, find the one-sided limit. x21x

In Exercises 5768, find the one-sided limit. 21x2

In Exercises 5768, find the one-sided limit. 1x2y

In Exercises 5768, find the one-sided limit. 1x2y

In Exercises 5768, find the one-sided limit. x2yx

In Exercises 5768, find the one-sided limit. 2yx2

In Exercises 5768, find the one-sided limit. yx2

In Exercises 5768, find the one-sided limit. x24

In Exercises 5768, find the one-sided limit. x24y

In Exercises 5768, find the one-sided limit. 24y

In Exercises 5768, find the one-sided limit. 4y0

Environment A utility company burns coal to generate electricity. The cost in dollars of removing of the air pollutants in the stack emissions is 4y0y

Find the cost of removing (a) 15%, (b) 50%, and (c) 90% of the pollutants. (d) Find the limit of as 70. The function is defined as shown. (a) Find (if it exists). (b) Can the function be defined at such that it is continuous at x 0? y0y

Let be a point on the parabola in the first quadrant. Consider the triangle formed by and the origin , and the triangle formed by and the origin. (a) Write the perimeter of each triangle in terms of (b) Let be the ratio of the perimeters of the two triangles, Complete the table. (c) Calculate 0yx

Let be a point on the parabola in the first quadrant. Consider the triangle formed by and the origin , and the triangle formed by and the origin. (a) Write the area of each triangle in terms of (b) Let be the ratio of the areas of the two triangles, Complete the table. (c) Calculate 0yx2

(a) Find the area of a regular hexagon inscribed in a circle of radius 1. How close is this area to that of the circle? (b) Find the area of an -sided regular polygon inscribed in a circle of radius 1. Write your answer as a function of (c) Complete the table. (d) What number does approach as gets larger and larger? Figure for 3 Figure for 4 yx2

Let be a point on the circle (a) What is the slope of the line joining and (b) Find an equation of the tangent line to the circle at (c) Let be another point on the circle in the first quadrant. Find the slope of the line joining and in terms of (d) Calculate How does this number relate to your answer in part (b)? x23

Let be a point on the circle (a) What is the slope of the line joining and (b) Find an equation of the tangent line to the circle at (c) Let be another point on the circle in the fourth quadrant. Find the slope of the line joining and in terms of (d) Calculate How does this number relate to your answer in part (b)? x23x

Find the values of the constants and such that 23x

Consider the function (a) Find the domain of (b) Use a graphing utility to graph the function. (c) Calculate . (d) Calculate 3x3

Determine all values of the constant such that the following function is continuous for all real numbers. 3x3x

Consider the graphs of the four functions and . For each given condition of the function which of the graphs could be the graph of (a) (b) is continuous at 2. (c) x3x

Sketch the graph of the function . (a) Evaluate and (b) Evaluate the limits , and . (c) Discuss the continuity of the function. 3x1

Sketch the graph of the function (a) Evaluate and (b) Evaluate the limits and . (c) Discuss the continuity of the function. 3x1

To escape Earths gravitational field, a rocket must be launched with an initial velocity called the escape velocity. A rocket launched from the surface of Earth has velocity (in miles per second) given by where is the initial velocity, is the distance from the rocket to the center of Earth, is the gravitational constant, is the mass of Earth, and is the radius of Earth (approximately 4000 miles). (a) Find the value of for which you obtain an infinite limit for as tends to zero. This value of is the escape velocity for Earth. (b) A rocket launched from the surface of the moon has velocity (in miles per second) given by Find the escape velocity for the moon. (c) A rocket launched from the surface of a planet has velocity (in miles per second) given by Find the escape velocity for this planet. Is the mass of this planet larger or smaller than that of Earth? (Assume that the mean density of this planet is the same as that of Earth.) x12

For positive numbers the pulse function is defined as where is the Heaviside function. (a) Sketch the graph of the pulse function. (b) Find the following limits: (i) (ii) (iii) (iv) (c) Discuss the continuity of the pulse function. (d) Why is called the unit pulse function? 12y

Let be a nonzero constant. Prove that if then Show by means of an example that must be nonzero. 12y