Textbook Question
Vertical tangent lines
b. Does the curve have any horizontal tangent lines? Explain.
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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 3.42b
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = 1/3x-1; a= 2
Verified step by step guidance1
Step 1: Identify the function and the point of tangency. The function given is \( f(x) = \frac{1}{3}x - 1 \) and the point of tangency is \( (a, f(a)) \) where \( a = 2 \).
Step 2: Calculate \( f(a) \) to find the y-coordinate of the point of tangency. Substitute \( a = 2 \) into the function: \( f(2) = \frac{1}{3}(2) - 1 \).
Step 3: Find the derivative of the function \( f(x) \) to determine the slope of the tangent line. The derivative \( f'(x) \) of \( f(x) = \frac{1}{3}x - 1 \) is \( f'(x) = \frac{1}{3} \).
Step 4: Evaluate the derivative at \( x = a \) to find the slope of the tangent line at the point. Since \( f'(x) = \frac{1}{3} \), the slope \( m \) at \( x = 2 \) is \( m = \frac{1}{3} \).
Step 5: Use the point-slope form of a line to write the equation of the tangent line. The point-slope form is \( y - f(a) = m(x - a) \). Substitute \( m = \frac{1}{3} \), \( a = 2 \), and \( f(a) \) from Step 2 into the equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function at a particular point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In practical terms, the derivative at a point gives the slope of the tangent line to the graph of the function at that point.
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Derivatives
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 is equal to the derivative of the function at that point. The equation of the tangent line can be expressed in point-slope form, which utilizes the slope and the coordinates of the point of tangency.
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05:13Slopes of Tangent Lines
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. It allows for a straightforward way to express the line based on its slope and a specific point.
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Guided course
3:56Slope-Intercept Form
Related Practice
Textbook Question
A race Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions θ(t) and ϕ(t), respectively, where 0≤t≤4 and t is measured in minutes (see figure). These angles are measured in radians, where θ=ϕ=0 represent the starting position and θ=ϕ=2π represent the finish position. The angular velocities of the runners are θ′(t) and ϕ′(t). <IMAGE>
b. Which runner has the greater average angular velocity?
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Textbook Question
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.
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Textbook Question
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
4x³ =y²(4−x); x=2 (cissoid of Diocles)
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Textbook Question
Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = (x + 5) / (x - 1); a = 3
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Textbook Question
City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.
b. How fast will the city be growing when it reaches a size of 38 mi²?
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