In Exercises 5-10 , find an equation for the tangent to the curve at the given point. Then sketc

In Exercises 5-10 , find an equation for the tangent to the...

Key Concepts

Point-Slope Form

The point-slope form of a linear equation, expressed as y - y? = m(x - x?), is used to find the equation of a line when a point (x?, y?) on the line and the slope m are known. This form is particularly useful for writing the equation of the tangent line after determining the slope via differentiation.

Graphical Analysis

Graphical analysis involves sketching the original function along with its tangent line to obtain a visual understanding of the behavior of the function. This aids in comprehending concepts such as local linearity, rates of change, and the approximation of nonlinear functions by linear ones.

Derivative

The derivative of a function represents the instantaneous rate of change of that function, or equivalently, the slope of the tangent line to the curve at a specific point. It is a foundational concept in calculus that enables the analysis of how functions change and is computed using rules such as the limit definition, power rule, product rule, and chain rule.

Tangent Line

A tangent line to a curve at a given point is the straight line that touches the curve at that point without crossing it in the local vicinity. This line serves as the best linear approximation to the curve near the point of tangency and its equation can be derived using the point-slope form once the slope (found via the derivative) is known.

Transcript

00:01 For this problem.

00:02 We want to find the tangent line to a given curve at a given point and then graft both of these lines, we are given the curve.

00:11 Why equals two times the square root of x and the point want to do this problem there? two things i mean you to remember the first is that the form of an equation line in slope intercept for is why equals, um packs plus be in which m is the slope of the line, and b is the y intercept.

00:34 The second thing you to know is that the first derivative of a curve at a given point is equal to the slope of that tent of the tangent line at that given point.

00:47 So, first, all this problem we want to find the slope of our tangent line, which is gonna be the derivative of why at the point 12 so to do this, we're gonna use their differentiation rules.

01:04 Um, burning is the constant room on the power.

01:07 So using the constant rule, we know that the two is just gonna come right out front and stay there.

01:13 And since it's a square root of x x is raised to the power of 1/2.

01:17 So using power world rules, we know to bring out the 1/2 in front and then we're gonna subtract one from 1/2.

01:26 We get x, the negative 1/2.

01:29 Thanks.

01:32 And then we want to plug in the points one to so we see that the two and the 1/2 will cancel toe one.

01:43 So we're left with one times one to the negative 1/2 which is the same as one times one over the square root of one.

01:56 So scared of one is one and 1/1 is also one and one times one is also one.

02:01 So we get to the slope is one.

02:03 Now i want to find the y intercept.

02:05 Be so so far we have that why equals one x plus be to find b.

02:18 You wanna plug in a point and unsolved for being.

02:21 Now we need a point that we know is on the tangent line...

Please give Ace some feedback