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Mathematically: y = mx +b, with m = 0 (constant function), or m =/= 0 (linear). Intuitively: discuss something that is responsive or not. Perhaps adding weight to a scale, before (or after) it hits the highest level on the indicator.

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A constant function does not vary with the independent variable: its value remains the same no matter what $x$ is (assuming $x$ is the independent variable). As a formula, a line can be described by $y = ax + b$. The special case of a constant function has $a = 0$, so the formula can be written succinctly as $y = b$. If you want to bring in the idea of slope,...

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The linear function is a pretty good first example: they are very fundamental and everything is simple. But rather than think in terms of the most fundamental thing to come next, I would encourage you to think in terms of typical examples that show the breadth of the concept. Linear function Polynomial of maybe second degree (not linear, but still fairly ...

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I think it is important to constantly connect implicit differentiation with the chain rule. At first, there might be examples where a third variable is the independent variable, and $x$ and $y$ are dependent variables. For example, find $\frac{dy}{dx}$ if $x(t)^2+y(t)^2=1$, so $2x\frac{dx}{dt}+2y\frac{dy}{dt}=0$. In this case, the coordinate variables are of ...

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I introduce implicit differentiation in two parts: How to differentiate dependent variables (i.e. $\frac{d}{dx} y = \frac{dy}{dx}$) If two functions are equal not just at a point, their derivatives are equal And then go through a bunch of examples including the squircle $x^4+y^4=1$ and plenty of just "find $\frac{d}{dx} \left(xy + x^2 + y^2\right)$&...

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I haven't taught calculus in a while and looking back, implicit differentiation is one of the (many) topics that I think I could have taught better. If I was teaching it now I would emphasize that the derivative is a local property, and show how by zooming in on a point on the graph we can find a region where $y$ is a function of $x$ (except when we can't, ...

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Without being rigourous, the equation $F(x,y)=0$ defines a relation between $y$ and $x$. To promote this relation to a function, some restrictions in the $(x,y)$-plane has to be imposed. Assuming that such conditions exists, you can say that the equation $F(x,y)=0$ is defining $y$ implicitly as a function of $x$. In some cases, we will be able to manipulate ...

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