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(a) The curve with equation $ y^2 = 5x^4 - x^2 $ is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point (1, 2).(b) Illustrate part (a) by graphing the curve and the tangent line on a common screen. (If your graphing device will graph implicitly defined curves, then use that capability. If not. you can still graph this curve by graphing its upper and lower halves separately.)

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00:38

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Oregon State University

Harvey Mudd College

Baylor University

Boston College

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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(a) The curve with equatio…

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a. Find an equation of the…

(a) The curve $ y = \mid x…

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(a) Usc a graphing utility…

In this problem were given a curve we are asked to find in part a the equation of tangent line at .12 in part, were asked to plot on the same graph, the function itself and the derivative of that function. So we know that tangent line will be from y. Minus y is equal to derivative, a given point x and multiplied by x, minus x, and so let's find derivative function. First, let's take derivative on both sides. We have 2 y times y prime, is equal to 20 x cubed minus 2 x. We know x, not a 1 that is given, so we can find what am at a given point plus pluck the given values, and so we have 4 times what prime is equal to 20 minus 2, and that is 18 from this. We can see that y prime will be 9 over 2 point all right if y prime is 9 over 2 point. It is this term we can find the equation of 10 lines. Y minus 2 is equal to 9 over 2 x. Minus 1 from the standard equation is 9 x over 2 minus 5, o 2 point now in part b, you can use any ling bunch of plotter or even your calculators. This is what we have. The function looks like this is our function itself y, and here is our point so 2, that is y axis, and that is the x axis, and this is the derivative of this function y prime- and that is the tangent point right here.

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