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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 41b
Chapter 3, Problem 41b

Slopes and Tangent Lines


b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope?

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To find the smallest slope on the curve, we need to determine the derivative of the function that defines the curve. The derivative represents the slope of the tangent line at any point on the curve.
Once we have the derivative, we need to find the critical points by setting the derivative equal to zero and solving for the variable. These points are where the slope could potentially be at a minimum or maximum.
After finding the critical points, we should evaluate the second derivative to determine the concavity of the function at these points. This will help us identify whether each critical point is a minimum, maximum, or a point of inflection.
The smallest slope will occur at the critical point where the second derivative is positive, indicating a local minimum. Evaluate the first derivative at this point to find the slope.
Finally, substitute the critical point back into the original function to find the corresponding point on the curve where this smallest slope occurs.

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Key Concepts

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

Derivative

The derivative of a function at a point provides the slope of the tangent line to the curve at that point. It is a fundamental tool in calculus for understanding how a function changes. To find the smallest slope on a curve, one must first determine the derivative of the function representing the curve.
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Derivatives

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are potential candidates for local minima, maxima, or points of inflection. To find the smallest slope, identify the critical points by setting the derivative equal to zero and solving for the variable.
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Critical Points

Second Derivative Test

The second derivative test helps determine the nature of critical points found from the first derivative. If the second derivative at a critical point is positive, the function has a local minimum there. This test is useful for confirming that a critical point corresponds to the smallest slope on the curve.
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The Second Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

Slopes and Tangent Lines


a. Horizontal tangent lines Find equations for the horizontal tangent lines to the curve y = x³ − 3x − 2. Also find equations for the lines that are perpendicular to these tangent lines at the points of tangency.

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Textbook Question

The number of gallons of water in a tank t minutes after the tank has started to drain is Q(t) = 200(30 - t)². How fast is the water running out at the end of 10 min? What is the average rate at which the water flows out during the first 10 min?

Textbook Question

Vehicular stopping distance Based on data from the U.S. Bureau of Public Roads, a model for the total stopping distance of a moving car in terms of its speed is s = 1.1v + 0.054v², where s is measured in ft and v in mph. The linear term 1.1v models the distance the car travels during the time the driver perceives a need to stop until the brakes are applied, and the quadratic term 0.054v² models the additional braking distance once they are applied. Find ds/dv at v = 35 and v = 70 mph, and interpret the meaning of the derivative.

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Textbook Question

Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.

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Textbook Question

Airplane takeoff Suppose that the distance an aircraft travels along a runway before takeoff is given by D = (10/9)t², where D is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 200 km/h. How long will it take to become airborne, and what distance will it travel in that time?

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Textbook Question

Find by implicit differentiation.

x² + xy + y² - 5x = 2