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Chapter 11: Q. 6 (page 859)

Let y = f(x). State the definition for the continuity of the function f at a point c in the domain of f .

Short Answer

Expert verified

The continuity or discontinuity of f at a point 'c' can be said only when 'c' is in the domain of

f.

Step by step solution

01

. Given information is 

A function y=f(x) .

02

. Checking continuity of function f.

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Most popular questions from this chapter

Let x1t, x2t, y1t, and y2tbe differentiable scalar functions. Prove that the dot product of the vector-valued functions role="math" localid="1649579098744" r1t=x1t,y1tand role="math" localid="1649579122624" r2t=x2t,y2tis a differentiable scalar function.

In Exercises 19–21 sketch the graph of a vector-valued function r(t)=x(t),y(t) with the specified properties. Be sure to indicate the direction of increasing values oft.

Domain

t∈ℝ,r2=1,1,r0=-3,3,r-2=-5,-5limt→-∞x(t)=∞,limt→-∞y(t)=-∞,andlimt→∞r(t)=0,0.

Let r1(f)and r2(t) be differentiable vector functions with three components each. Prove that

ddtr1(t)-r2(t)=r1'(t)-r2(t)+r1(t)-r2'(t)∣(This is Theorem 11.11 (c).)

Annie is conscious of tidal currents when she is sea kayaking. This activity can be tricky in an area south-southwest of Cattle Point on San Juan Island in Washington State. Annie is planning a trip through that area and finds that the velocity of the current changes with time and can be expressed by the vector function

0.4cos⁡π(t−8)6,−1.1cos⁡π(t−11)6,

where t is measured in hours after midnight, speeds are given in knots and point due north.

(a) What is the velocity of the current at 8:00 a.m.?

(b) What is the velocity of the current at 11:00 a.m.?

(c) Annie needs to paddle through here heading southeast, 135 degrees from north. She wants the current to push her. What is the best time for her to pass this point? (Hint: Find the dot product of the given vector function with a vector in the direction of Annie’s travel, and determine when the result is maximized.)

For each of the vector-valued functions, find the unit tangent vector .

r(t)=(t,t2)
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