Textbook Question
5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = √7x-1
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 19a
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = √(3x + 3); P(2,3)
Verified step by step guidance1
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 f(x) = \(\sqrt{3x + 3}\) and the point P(2, 3). Here, a = 2, and f(a) = f(2) = 3, which matches the y-coordinate of point P.
Step 3: Substitute a = 2 into the derivative definition: f'(2) = \(\lim\)_{h \(\to\) 0} \(\frac{f(2+h) - f(2)}{h}\).
Step 4: Calculate f(2+h) by substituting into the function: f(2+h) = \(\sqrt{3(2+h) + 3}\) = \(\sqrt{6 + 3h + 3}\) = \(\sqrt{9 + 3h}\).
Step 5: Substitute f(2+h) and f(2) into the limit expression: f'(2) = \(\lim\)_{h \(\to\) 0} \(\frac{\sqrt{9 + 3h}\) - 3}{h}. Simplify and evaluate this 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) will provide the slope of the tangent line at point P.
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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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Related Practice
Textbook Question
5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.
y = √x²+1
Textbook Question
Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
On what intervals is the speed increasing?
f(t) = 18t - 3t2; 0 ≤ t ≤ 8
Textbook Question
Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
Determine the velocity and acceleration of the object at t = 1.
f(t) = 2t3 - 21t2 + 60t; 0 ≤ t ≤ 6
Textbook Question
Position, velocity, and acceleration Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
a. Graph the position function.
1
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Textbook Question
Position, velocity, and acceleration Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left?
f(t) = 18t-3t²; 0 ≤ t ≤ 8