Let f(x)=x^3+3 x^2-2 x+1 and g(x)=x^3+3 x^2-2 x-4 (a) Show...

Let $f(x)=x^{3}+3 x^{2}-2 x+1$ and $g(x)=x^{3}+3 x^{2}-2 x-4$ (a) Show that the derivatives of $f(x)$ and $g(x)$ are the same. (b) Use the graphs of $f(x)$ and $g(x)$ to explain why their derivatives are the same. (c) Are there other functions which share the same derivative as $f(x)$ and $g(x) ?$

Let $f(x)=x^{3}+3 x^{2}-2 x+1$ and $g(x)=x^{3}+3 x^{2}-2 x-4$
(a) Show that the derivatives of $f(x)$ and $g(x)$ are the
same.
(b) Use the graphs of $f(x)$ and $g(x)$ to explain why their derivatives are the same.
(c) Are there other functions which share the same derivative as $f(x)$ and $g(x) ?$

Key Concepts

Differentiation of Functions

Differentiation is the process of finding the derivative of a function, which measures the rate at which the function value changes with respect to changes in its input. In the context of the problem, both f(x) and g(x) are polynomial functions whose derivatives are found by applying standard differentiation rules (like the power rule) term by term.

Constant Difference (Vertical Translation) Theorem

The constant difference theorem states that if two functions differ only by a constant, then their derivatives are identical. This is because the derivative of a constant is zero, and therefore any vertical shift (addition or subtraction of a constant) does not affect the rate of change of the function.

Graphical Interpretation of Derivatives

Graphically, the derivative represents the slope of the tangent line at each point on a function's curve. When two functions have graphs that are vertical translations of each other (shifted up or down by a constant), the slopes at corresponding points remain the same, thereby resulting in identical derivatives. This concept explains why f(x) and g(x) have the same derivative despite differing by a constant term.

Transcript

00:01 You're given two different functions.

00:03 F of x equals x q plus 3x squared minus 2x plus 1 and g of x is given to you as the same thing except instead of plus 1 you have a minus 4.

00:26 So in part a when you look at the derivative of each of those what you're going to see is you know you move that exponent in front and you'll multiply and you subtract one from your exponents.

00:39 I guess i didn't have to write that to the first power.

00:43 What you'll see is they're identical.

00:46 3x squared plus 6x minus 2.

00:51 So there's your answer to part a.

00:54 Circle that.

00:55 And so what we're examining is if you were to look at these two graphs individually, the graph of f and the graph of g, you'd have the same function you know it goes up and back down and then back up except one of them so let's say that this is the other one is shifted down you know instead of going up one it shifted down for your y intercept so what we're trying to get at for letter b if you look at the two graphs is the slope i guess, yeah, the slope at each x value is the same.

01:47 Now, you don't want to say the slope at each point is the same because this point is corresponding to this point, which would be the same x value.

01:58 So you want to say the slope at each x value is the same throughout, which is what we're trying to find when we find the derivative of each.

02:06 So i'll circle that.

02:07 There might be a better phrase of that, but i'm okay with my students do things like that...

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