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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 3.1.20a
Chapter 3, Problem 3.1.20a

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = 2/√x; P(4,1)

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1
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 \( f(x) = \frac{2}{\sqrt{x}} \) and the point P is (4, 1). We need to find \( f'(4) \).
Step 3: Substitute \( a = 4 \) into the derivative definition: \( f'(4) = \lim_{h \to 0} \frac{f(4+h) - f(4)}{h} \).
Step 4: Calculate \( f(4+h) \) and \( f(4) \). We have \( f(4) = \frac{2}{\sqrt{4}} = 1 \). For \( f(4+h) \), substitute into the function: \( f(4+h) = \frac{2}{\sqrt{4+h}} \).
Step 5: Substitute \( f(4+h) \) and \( f(4) \) into the limit expression: \( f'(4) = \lim_{h \to 0} \frac{\frac{2}{\sqrt{4+h}} - 1}{h} \). Simplify the expression and evaluate the limit to find the slope of the tangent line.

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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) = 2/√x 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, as it formalizes the concept of instantaneous rate of change.
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Definition of the Definite Integral
Related Practice
Textbook Question

Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = -3x2 - 5x + 1; P(1,-7)

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Textbook Question

Volume of a torus The volume of a torus (doughnut or bagel) with an inner radius of a and an outer radius of b is V=π²(b+a)(b−a)²/4.

a. Find db/da for a torus with a volume of 64π².

Textbook Question

79–82. {Use of Tech} Visualizing tangent and normal lines <IMAGE>

a. Determine an equation of the tangent line and the normal line at the given point (x0, y0) on the following curves. (See instructions for Exercises 73–78.)

x⁴ = 2x²+2y²; (x0, y0)=(2, 2) (kampyle of Eudoxus)

Textbook Question

A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e-t cos t), for t ≥ 0.

Determine her velocity at t = 1 and t = 3. 

Textbook Question

45–50. Tangent lines Carry out the following steps. <IMAGE>

a. Verify that the given point lies on the curve.

(x²+y²)²=25/4 xy²; (1, 2)

Textbook Question

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

f(x) = 2x + 1; P(0,1)