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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 31b
Chapter 3, Problem 31b

Equations of tangent lines by definition (2)
b. Determine an equation of the tangent line at P.
f(x) = √x+3; P (1,2)

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Step 1: Understand that the equation of a tangent line to a curve at a given point is given by the formula: y - y_1 = m(x - x_1), where m is the slope of the tangent line, and (x_1, y_1) is the point of tangency.
Step 2: Identify the function f(x) = \(\sqrt{x}\) + 3 and the point P(1, 2). Here, x_1 = 1 and y_1 = 2.
Step 3: To find the slope m of the tangent line, calculate the derivative of the function f(x). The derivative f'(x) represents the slope of the tangent line at any point x.
Step 4: Differentiate f(x) = \(\sqrt{x}\) + 3. The derivative f'(x) = \(\frac{1}{2\sqrt{x}\)}. This is because the derivative of \(\sqrt{x}\) is \(\frac{1}{2\sqrt{x}\)}, and the derivative of a constant is 0.
Step 5: Evaluate the derivative at x = 1 to find the slope of the tangent line at P. Substitute x = 1 into f'(x) to get m = f'(1).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Line Definition

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. Mathematically, it represents the instantaneous rate of change of the function at that point, which is equivalent to the derivative of the function evaluated at that point.
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Slopes of Tangent Lines

Derivative

The derivative of a function at a point quantifies how the function's output changes as its input changes. It is calculated as the limit of the average rate of change of the function as the interval approaches zero. For the function f(x) = √(x + 3), the derivative will provide the slope of the tangent line at point P.
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Derivatives

Point-Slope Form

The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for writing the equation of a tangent line once the slope (derivative) and the point of tangency are known.
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Slope-Intercept Form
Related Practice
Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = √(x + 3); P (1,2)

Textbook Question

Calculate the derivative of the following functions.

y = sec(3x+1)

Textbook Question

Consider the following cost functions.

c. Interpret the values obtained in part (b).

C(x) = 500+0.02x, 0≤x≤2000, a=1000

Textbook Question

Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function D(p)=40−2pD(p) = 40-2p , where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

a. According to the model, how many DVDs can be sold in a day at a price of \$10?

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Textbook Question

Find and simplify the derivative of the following functions.

h(x) = (x − 1)(x3+ x2 + x+1)

Textbook Question

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

Find the elasticity function for this demand function.