Question Video: Vector-Valued Functions | Nagwa Question Video: Vector-Valued Functions | Nagwa

Question Video: Vector-Valued Functions Mathematics • Higher Education

For the given function 𝐫(𝑑) = 2𝑒^(𝑑 βˆ’ 1) 𝐒 + cos (π‘‘πœ‹)𝐣 βˆ’ ln (𝑑)𝐀, evaluate 𝐫(1).

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Video Transcript

For the given function 𝐫 of 𝑑 equals two 𝑒 to the power of 𝑑 minus one 𝐒 plus cos of π‘‘πœ‹ 𝐣 minus the natural log of 𝑑 𝐀, evaluate 𝐫 of one.

In this question, we’ve been given a vector-valued function. It’s a function whose range is a vector or set of vectors and whose domain is a subset of the real numbers. We’re being asked to evaluate 𝐫 of one. In other words, we need to let 𝑑 be equal to one. So we replace 𝑑 with one in each part of our function.

So the component for 𝐒 becomes two 𝑒 to the power of one minus one. The component of 𝐣 is cos of one πœ‹. And the component for 𝐀 is negative the natural log of one. Now of course, two 𝑒 to the power of one minus one is two 𝑒 to the power of zero. And cos of one πœ‹ is equal to cos of πœ‹. So we have two 𝑒 to the power of zero 𝐒 plus cos of πœ‹ 𝐣 minus the natural log of one 𝐀.

But of course, anything to the power of zero is one. And so the component of 𝐒 here is two. We know cos of πœ‹ to be equal to negative one. So the component for 𝐣 is negative one. And we can write 𝐫 of one as two 𝐒 minus 𝐣. But we also know that the natural log of one is zero. So 𝐫 of one is two 𝐒 minus one 𝐣 plus zero 𝐀, which is simply two 𝐒 minus 𝐣.

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