Chapter 2: Q. 94 (page 186)

Use Problem 93 to prove that a linear function is its own tangent line at every point. In other words, show that if $f (x) = m x + b$ is any linear function, then the tangent line to $f$ at any point $x = c$ is given by $y = m x + b$ .

Short Answer

Expert verified

Ans:

$y - f (c) = f^{'} (c) [x - c] y - (m c + b) = m [x - c] y - m c - b = m x - m c y = m x - m c + m c + b y = m x + b$

Step by step solution

Step 1. Given information:

Consider the function:

$f (x) = m x + b$

Here the objective is to show that the equation of the tangent line at any point $x = c$ is $y = m x + b$ .

Step 2. Solving the equation:

$f^{'} (x) = \lim_{z \to 4} \frac{f (z) - f (x)}{z - x} = \lim_{z \to x} \frac{(m z + b) - (m x + b)}{z - x} = \lim_{z \to x} \frac{m (z - x)}{z - x} = \lim_{z \to x} m = m$

Hence the slope of the linear function is constant and it is $f' (x) = m$

At any point x=c the slope of the function is $f' (c) = m$

At x=c, the value of the function is

$f (c) = m c + b$

Step 3. Finding the equation of the tangent line:

The eq of the tangent line will be,

$y - f (c) = f^{'} (c) [x - c] y - (m c + b) = m [x - c] y - m c - b = m x - m c y = m x - m c + m c + b y = m x + b$

Hence the eq of the tangent is same as linear function.

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Use Problem 93 to prove that a linear function is its own tangent line at every point. In other words, show that if $f (x) = m x + b$ is any linear function, then the tangent line to $f$ at any point $x = c$ is given by $y = m x + b$ .

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the equation:

Step 3. Finding the equation of the tangent line:

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Use Problem 93 to prove that a linear function is its own tangent line at every point. In other words, show that if f(x)=mx+bis any linear function, then the tangent line tofat any point x=cis given by y=mx+b.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the equation:

Step 3. Finding the equation of the tangent line:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Geometry

Applied Mathematics

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use Problem 93 to prove that a linear function is its own tangent line at every point. In other words, show that if $f (x) = m x + b$ is any linear function, then the tangent line to $f$ at any point $x = c$ is given by $y = m x + b$ .