Let f(x)=x^2 . For what value(s) of c does the tangent line to the graph of f at (c, f(c)) pass

Let f(x)=x^2 . For what value(s) of c does the tangent line...

Key Concepts

Tangent Line

A tangent line to a curve at a given point touches the curve at that point and has a slope equal to the derivative at that point. It serves as the best linear approximation to the function near that point and is foundational in differential calculus.

Derivative

The derivative of a function measures its instantaneous rate of change and provides the slope of the tangent line at any specific point on a curve. This concept is essential for finding how a function behaves locally and is used to determine the slope required for constructing tangent lines.

Point-Slope Form

The point-slope form is a method to express the equation of a straight line using a known point on the line and its slope. It is particularly useful for writing the equation of a tangent line when the slope (from the derivative) and the point of tangency are known.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation that often arises when setting up conditions based on the slopes or intersections of functions. Solving a quadratic equation can determine unknown quantities that satisfy the geometric or algebraic conditions established in a problem.

Transcript

00:01 Okay, so we're going to be finding a tangent line to this function at the c value equaling 1.

00:06 So what we want to do is we want to figure out first what f of 1 is equal to.

00:11 So that's going to be 1 squared, which is 1 minus 3 times 5 times 1, which is 5 minus 1 cubed, which is 1.

00:18 So this is negative 2 times 4, which is equal to negative 8.

00:24 So we're looking at the ordered pair 1 comma 8, and that's going to be important since the equation, for a tangent line is y minus y1 is equal to f prime of x1 times x minus x1 where y 1 and x1 are this ordered pair here x1 is the x coordinate and y 1 is the y coordinate so really all we need to find now is our derivative function in what the derivative value is at x is equal to 1 so let's go ahead and find f prime of x if we look at our function f of x it's two terms multiplied together or two functions multiplied together and so we can use a product rule here which is the derivative of this first function x squared minus 3 and the derivative of that is just 2x i use the power rule on x squared and then minus 3 3 is a constant so that's going to go to zero and then we have multiplied by the second term so 5x minus x cubed and then plus the first which is x squared minus 3 and then times the derivative of 5x minus x cubed.

01:28 So that's going to be 5 minus 3x squared.

01:32 And now that we have our derivative function, we don't need to multiply these factors together.

01:37 We just need to find the value of our derivative function at once.

01:41 So let's just plug in 1 for x...

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