Let f(x)=x^2+6 x a. Find the derivative f^' of f b. Find the point on the graph of f where the t

step 1 We just have a tangent line problem. And any time you have a tangent line, you need a point, whi

Let f(x)=x^2+6 x a. Find the derivative f^' of f b. Find the...

Let $f(x)=x^{2}+6 x$ a. Find the derivative $f^{\prime}$ of $f$ b. Find the point on the graph of $f$ where the tangent line to the curve is horizontal. Hint: Find the value of $x$ for which $f^{\prime}(x)=0$ c. Sketch the graph of $f$ and the tangent line to the curve at the point found in part (b).

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes at each point. It is a fundamental concept in calculus that represents the slope of the tangent line to the function's graph at any given point, providing insights into the function’s behavior, such as increasing or decreasing trends.

Power Rule

The power rule is a basic differentiation technique used to compute the derivative of polynomial terms. It states that for any term of the form x^n, the derivative is n*x^(n-1), which simplifies the process of finding derivatives for functions involving powers of x.

Tangent Line

A tangent line to a curve at a specific point is a straight line that touches the curve only at that point and has the same slope as the curve at that point. The tangent line is used to approximate the function's behavior near the point of tangency and is directly determined by the derivative at that point.

Horizontal Tangent

A horizontal tangent line occurs at points where the derivative of a function is zero. This indicates that at those points, the instantaneous rate of change of the function is zero, which often corresponds to local extrema or flat points on the graph.

Graphing Functions

Graphing functions involves plotting the function's curve based on its algebraic form, analyzing key features like intercepts, turning points, and asymptotic behavior, and adding elements such as tangent lines to illustrate specific properties of the function, all of which help in understanding the overall behavior of the function.

Transcript

00:01 We just have a tangent line problem.

00:03 And any time you have a tangent line, you need a point, which they give you the point in part b.

00:09 Actually, they don't give you the point.

00:15 We have to find the point, and we also need to find the slope.

00:18 So in part a, they give us the function is equal to x squared plus 6x.

00:30 So in order to find the derivative f prime, we need to take.

00:35 The limit as h approaches 0 of f of x plus h minus f of x all over h.

00:44 Now, re -looking at that problem, that h, we can't do anything.

00:53 We already know that f of x is equal to x squared plus 6x.

00:58 And what we need to do is figure out what f of x plus h is.

01:01 Well, we figure that out by replacing both of these xs with x plus h.

01:07 So it's still squared, and then six, and replace the.

01:10 This x with x plus h so as you go to simplify i'll go ahead and simplify all this uh we would foil out because it's x plus h times x plus h so there's two xh in there plus h squared also need to distribute that six in there and then distribute the minus and what you'll notice hopefully you notice you can show all the work if you need to the x squared cancels six x cancels and every other piece has at least one h in it that we could cancel out with this h in the denominator.

01:49 So the derivative for part a is literally just 2x.

01:58 And if we plug in 0 for this h, we get zero so you don't have to write that down plus 6.

02:05 I don't know if you can see.

02:06 There's a 6 right there with that.

02:08 So there's the equation of the derivative of f of x, 2x plus 6...

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