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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 31a
Chapter 3, Problem 31a

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)

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Step 1: Recall the definition of the derivative as the slope of the tangent line at a point. The derivative of a function f at a point x = a is given by the limit: \( f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).
Step 2: Identify the function and the point of tangency. Here, the function is \( f(x) = \sqrt{x + 3} \) and the point P is (1, 2).
Step 3: Substitute the point of tangency into the derivative definition. We need to find \( f'(1) \), so substitute \( a = 1 \) into the limit: \( f'(1) = \lim_{h \to 0} \frac{\sqrt{1+h+3} - \sqrt{1+3}}{h} \).
Step 4: Simplify the expression inside the limit. This becomes \( \lim_{h \to 0} \frac{\sqrt{h+4} - 2}{h} \). To simplify further, multiply the numerator and the denominator by the conjugate of the numerator: \( \frac{\sqrt{h+4} - 2}{h} \times \frac{\sqrt{h+4} + 2}{\sqrt{h+4} + 2} \).
Step 5: Simplify the resulting expression. The numerator becomes \( (\sqrt{h+4})^2 - 2^2 = h + 4 - 4 = h \), so the expression simplifies to \( \lim_{h \to 0} \frac{h}{h(\sqrt{h+4} + 2)} \). Cancel \( h \) in the numerator and denominator, and evaluate the limit as \( h \to 0 \).

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Key Concepts

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

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is crucial for understanding how the function behaves locally.
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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 defined as the limit of the average rate of change of the function as the interval approaches zero. In this context, finding the derivative of f(x) = √(x + 3) will provide the slope of the tangent line at point P.
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Derivatives

Limit Definition of Derivative

The limit definition of the derivative states that the derivative f'(a) at a point a is the limit of the difference quotient as h approaches zero: f'(a) = lim(h→0) [(f(a + h) - f(a)) / h]. This definition is fundamental for calculating the slope of the tangent line using the function's values around the point of interest.
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Definition of the Definite Integral
Related Practice
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?

1
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Textbook Question

Equations of tangent lines by definition (2)

b. Determine an equation of the tangent line at P.

f(x) = √x+3; P (1,2)

Textbook Question

Find and simplify the derivative of the following functions.

h(x) = (x − 1)(x3+ x2 + x+1)

Textbook Question

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = 500+0.02x, 0≤x≤2000, a=1000