find an equation for the line tangent to the curve at the given point. y=(x^3+1)/(2 x), x=1

Name: Problem 27, Chapter 3: Calculus: Graphical, Numerical, Algebraic
Uploaded: 2021-08-27T15:36:55-08:00
Duration: 1 min 57 s
Channel: Adrian Co
Description: find an equation for the line tangent to the curve at the given point. y=(x^3+1)/(2 x), x=1

step 1 So our objective here is to get the equation of a tangent line at this particular x value for th

Find an equation for the line tangent to the curve at the...

Key Concepts

Quotient Rule

The quotient rule is a formula used in calculus to differentiate functions that are expressed as the ratio of two differentiable functions. It states that the derivative of a quotient is given by dividing the difference of the derivative of the numerator times the denominator and the numerator times the derivative of the denominator, by the square of the denominator. This rule is essential when handling functions presented as fractions.

Point-Slope Form

The point-slope form is a way of expressing the equation of a line when a point on the line and its slope are known. It is given by the formula y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope. This form is particularly useful in situations involving tangent lines where the slope is obtained from the derivative and the point of tangency is known.

Tangent Line

A tangent line to a curve at a given point is the straight line that locally touches the curve and has the same instantaneous direction (slope) as the curve at that point. It represents the line that best approximates the curve near that point and is used in various applications such as linear approximations and instantaneous rate of change.

Derivative

The derivative of a function at a given point measures the rate at which the function's value changes with respect to changes in the input variable. In geometric terms, the derivative at a point gives the slope of the tangent line to the curve at that point, making it a fundamental tool for analyzing the behavior of functions.

Transcript

Please give Ace some feedback