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Nov 16, 2022 · In this section discuss how the gradient vector can be used to find tangent planes to a much more general function than in the previous section. We will also define the normal line and discuss how the gradient vector can be used to find the equation of the normal line.
- Practice Problems
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- Assignment Problems
Here is a set of assignement problems (for use by...
- Practice Problems
- Tangent Lines
- Rates of Change
- Velocity Problem
- Change of Notation
The first problem that we’re going to take a look at is the tangent line problem. Before getting into this problem it would probably be best to define a tangent line. A tangent line to the function f(x)f(x) at the point x=ax=ais a line that just touches the graph of the function at the point in question and is “parallel” (in some way) to the graph ...
The next problem that we need to look at is the rate of change problem. As mentioned earlier, this will turn out to be one of the most important concepts that we will look at throughout this course. Here we are going to consider a function, f(x)f(x), that represents some quantity that varies as xx varies. For instance, maybe f(x)f(x) represents the...
Let’s briefly look at the velocity problem. Many calculus books will treat this as its own problem. We however, like to think of this as a special case of the rate of change problem. In the velocity problem we are given a position function of an object, f(t)f(t), that gives the position of an object at time tt. Then to compute the instantaneous vel...
There is one last thing that we need to do in this section before we move on. The main point of this section was to introduce us to a couple of key concepts and ideas that we will see throughout the first portion of this course as well as get us started down the path towards limits. Before we move into limits officially let’s go back and do a littl...
Aug 29, 2023 · Solution. Use formula ( [eqn:tangentline]) with a = 0 and f(x) = x3. Then f(a) = f(0) = 03 = 0. The derivative of f(x) = x3 is f ′ (x) = 3x2, so f ′ (a) = f ′ (0) = 3(0)2 = 0. Hence, the equation of the tangent line is y − 0 = 0(x − 0), which is y = 0. In other words, the tangent line is the x -axis itself.
Describe the tangent problem and how it led to the idea of a derivative; Explain how the idea of a limit is involved in solving the tangent problem; Recognize a tangent to a curve at a point as the limit of secant lines; Identify instantaneous velocity as the limit of average velocity over a small time interval
Feb 22, 2021 · Calculate the first derivative of f (x). Plug the ordered pair into the derivative to find the slope at that point. Substitute both the point and the slope from steps 1 and 3 into point-slope form to find the equation for the tangent line.
Sep 15, 2024 · Learning Objectives. Relate the rate of change of a function to the slope of a secant line. Describe the concept and process of approximating the tangent line to a function at a given point. Recognize a tangent to a curve at a point as the limit of secant lines.
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Aug 17, 2024 · Determine the equation of a plane tangent to a given surface at a point. Use the tangent plane to approximate a function of two variables at a point. Explain when a function of two variables is differentiable. Use the total differential to approximate the change in a function of two variables.
