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A curve passes through the point (0, 5) and has the property that the slope of the curve at every point $ P $ is twice the $ y- $ coordinate of $ P. $ What is the equation of the curve?
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00:57
Wen Zheng
00:35
Amrita Bhasin
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 8
Exponential Growth and Decay
Derivatives
Differentiation
Missouri State University
Campbell University
Harvey Mudd College
Idaho State University
Lectures
04:40
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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in this problem. We're told that we have a curve whose slope is equal to twice the y coordinate. And you can think of slope as the derivative D Y D X, and then twice the y coordinate would be two times why. So when we see that equation, we might think about the main idea from this section where if you have a differential equation of the form, do I d t equals K times. Why it's solution will be this exponential growth and decay formula. So what that tells us is that we have something whose solution will be, Why equals? Why not times e to the K t. And in this case, que is too. And we're using X instead of t. Okay, so now what we need to know is the value of why not and that will complete are figuring out the model or the equation. So let's use the ordered pair that was given, we know goes through the 0.5 so we can substitute five and for the y value and zero in for the X value. And we have five equals. Why not times e to the two time zero So that would be five equals. Why not tons? Each of the zero each of the zero is one so five equals. Why not times one. So now we realize that why not? Is five. So if you substitute that in, you have the equation y equals five times e to the two x.
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