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(a) Sketch the graph of $ y = e^x $ as a curve in $ \mathbb{R}^2 $.(b) Sketch the graph of $ y = e^x $ as a surface in $ \mathbb{R}^3 $.(c) Describe and sketch the surface $ z = e^x $.

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02:43

Wen Zheng

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 6

Cylinders and Quadric Surfaces

Vectors

Missouri State University

Campbell University

Oregon State University

Idaho State University

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

07:58

$\begin{array}{l}{\text { …

00:29

Sketch the surface in 3 -s…

Okay, So what we see here is we first want to start by graphing. Why equals e to the X and we want to do it just in the are too. So that's the plane. Just that just involves X and y. So this is the graph that we see Onda were pretty familiar with the graph of y equals e to the X Because it's this typical exponential equation. Then we go into three dimensions and we want to know Well, what does it look like if we have y equals e to the X in this dimension in three dimensions? What we see it follows a very similar curve. If we were to just make this flat, it looks like the exact same curve that we saw before. Um, it would look something like this. However, the difference is if we move about in the third dimension, we see that it has a depth that goes as faras egos. So no matter what Z is, we have this y equals e to the X shape of the graph. But then it also has a third dimension that z next. What we want to dio is graph y equal actually easy equals. Eat the X So me grab Z equals easy X. We're going to see something very similar. Um, the only difference is now, instead of the width of the graph being related Thio Z Now the width of the graph is related to why is we see, no matter how far, why goes we have this e to the X looking graph, but in this case it's a function of Z, which is why it's going up here going down here. This is changing in with regards to Z as X increases or decreases, and then the width of the graph is a result of why.

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