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Continuity and limits of functions

Please refer to the attached file to view the complete questions.
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Question 1

Figure 1.1 y = f(x) = (2x+4)2 - (2x - 4)2
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Apply the slope predictor formula to find the slope of the line tangent to Figure 1.1. Then write the equation of the line tangent to the graph of f at the point (3, f(3)).

Question 2

Find all points on the curve y = (x + 4)(x - 5) at which the tangent line is horizontal.

Question 3

Figure 3.1 100^/¯2 ft/sec
Figure 3.2 y = x - ( x 25 )2
Suppose that a projectile is fired at an angle of 45 degrees from the horizontal. Its initial position is the origin in the xy-plane, and its initial velocity is Figure 3.1. Then its trajectory will be the part of the parabola Figure 3.2 for which y ≥ 0. a) How far does the projectile travel (horizontally) before it hits the ground? b) What is the maximum height above the ground that the projectile attains?

Question 4

Evaluate limx => 16 ^/¯x-4 x-16

Question 5

Figure 5.1 f(x) = 4 ^/¯x+8
Given Figure 5.1, use the four-step process to find a slopepredictor function m(x). Then write an equation for the line tangent to the curve at the point x = 8.

Question 6

Use one-sided limits to find the limit or determine that the limit does not exist. limx => 4 16-x2 4-x

Question 7
Find the trigonometric limit: limx => 0 sin3x 2x

Question 8

Use the Squeeze Law of limits to find the limit. limx =>0 x2 sin2 10x

Question 9

Figure 9.1 h(x) = x-9 |x-9|
Given Figure 9.1, tell where h is continuous. (Give your answer in interval form.)

Question 10

Figure 10.1 f(x) = x-4 x2-16
Given Figure 10.1, find all points where f is not defined (and therefore not continuous). For each such point, tell whether or not the discontinuity is removable.

Question 11

Figure 11.1 f(x) = { c2-x2if x<0 ccosx if x>0
Given Figure 11.1, find a value for c so that f(x) is continuous for all x.

Question 12

Figure 12.1 Given Figure 12.1, tell where f is continuous. (Give your answer in interval notation.)

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Solution Preview

Following is the text part of the solution. Please see the attached file for complete solution. Equations, diagrams, graphs and special characters will not appear correctly here. Thank you for using Brainmass.
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Question 1 Figure 1.1 y = f(x) = (2x+4)2 - (2x - 4)2
. Apply the slope predictor formula to find the slope of the line tangent to Figure 1.1. m =[ f(a+h) - f(a)]/hf(a+h) = [2(a+h) + 4]2 - [2(a+h) - 4]2 = 32(a+h)f(a) = [2a + 4]2 - [2a - 4]2 = 32am =[ f(a+h) - f(a)]/h = [32(a+h) - 32a]/h = 32Then write the equation of the line tangent to the graph of f at the point (3, f(3)). Slope is constant for all x.Slope at = 32Note: given function is (2x+4)2 - (2x-4)2 = 32 x which is a straight line with a slope of 32.Question 2 Find all points on the curve y = (x + 4)(x - 5) at which the tangent line is horizontal. y = x2 - x -20dy/dx = 2x - 1If the tangent line is horizontal, dy/dx = 0i.e. 2x - 1 = 0x = 0.5Tangent line is horizontal at x = 1/2Question 3 Figure 3.1 100^/¯2 ft/sec
Figure 3.2 y = x - ( x 25 )2
Suppose that a projectile is fired at an angle of 45 degrees from the ...

Solution Summary

Solutions to twelve problems involving functions, continuity, finding slope using predictor formula, tangent line to a curve, trajectory of a projectile, finding limits, finding limits using squeeze law and continuity of functions are provided. I have provided complete and detailed answer to all the questions. Solutions are in a 7-page word document.

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