1. Limits and Continuity

What does it mean for a function to be continuous on an interval?

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={2x if x<1

x^2+3x if x≥1; a=1

Complete the following steps for each function.

c. State the interval(s) of continuity.

f(x)={x^3+4x+1 if x≤0

2x^3 if x>0; a=0

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(x)=√25−x^2

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(x)=√x^2−3x+2

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(t)=(t^2−1)^3/2

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(x)=(2x−3)^2/3

Determine the interval(s) on which the following functions are continuous. At which finite endpoints of the intervals of continuity is f continuous from the left or continuous from the right? 

f(z)=(z−1)^3/4

Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 

f(x)=csc x;lim x→π/4f (x);lim x→2π^− f(x)

Determine the interval(s) on which the following functions are continuous; then analyze the given limits. 

f(x)=1+sin x / cos x; limx→π/2^− f(x); lim x→4π/3 f(x)

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car t hours after 7 a.m. on Friday morning, and let g(t) be your distance from the car t hours after 7 a.m. on Sunday morning.

a. Evaluate f(0), f(2), g(0), and g(2).

Suppose you park your car at a trailhead in a national park and begin a 2-hr hike to a lake at 7 A.M. on a Friday morning. On Sunday morning, you leave the lake at 7 A.M. and start the 2-hr hike back to your car. Assume the lake is 3 mi from your car. Let f(t) be your distance from the car t hours after 7 a.m. on Friday morning, and let g(t) be your distance from the car t hours after 7 a.m. on Sunday morning.

b. Let h(t)=f(t)−g(t). Find h(0) and h(2).

Let g(x)= {1 if x≥0

−1 if x<0.

a. Write a formula for |g(x)|.

Use the graph of $f$ in the figure to determine the values of $x$ in the interval $(-3,5)$ at which f fails to be continuous. Justify your answers using the continuity checklist.

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Determine the intervals of continuity for the parking cost function c introduced at the outset of this section (see figure). Consider 0≤t≤60. <FIGURE>