Domanda del libro di testo
For what value or values of the constant m, if any, is
ƒ(x) = { sin 2x, x ≤ 0
{ mx, x > 0
a. continuous at x = 0?
b. differentiable at x = 0?
Give reasons for your answers.
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Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 3, Problema 3.7.43
The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.
Guida verificata passo dopo passo1
To find the slope of the curve at a given point, we need to find the derivative of the curve equation with respect to x. The given equation is y⁴ = y² - x².
Differentiate both sides of the equation with respect to x. Use implicit differentiation: d/dx(y⁴) = d/dx(y² - x²).
Apply the chain rule to differentiate y⁴ and y²: 4y³(dy/dx) = 2y(dy/dx) - 2x.
Rearrange the equation to solve for dy/dx: dy/dx = (2x) / (4y³ - 2y).
Substitute the coordinates of the given points into the derivative to find the slope at each point. For the point (√3/4, √3/2), substitute x = √3/4 and y = √3/2. For the point (√3/4, 1/2), substitute x = √3/4 and y = 1/2.
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Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not isolated. In this case, the equation y⁴ = y² - x² involves both x and y, making it necessary to apply implicit differentiation to find the slope of the curve at specific points. This method allows us to differentiate both sides of the equation with respect to x, treating y as a function of x.
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Percorso guidato
05:14
Finding The Implicit Derivative
Slope of a Curve
The slope of a curve at a given point is defined as the derivative of the function at that point. It represents the rate of change of y with respect to x. For the curve defined by the equation y⁴ = y² - x², finding the slope at the specified points involves evaluating the derivative obtained from implicit differentiation and substituting the coordinates of the points into this derivative.
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05:45Understanding Slope Fields
Critical Points
Critical points are values of x where the derivative of a function is either zero or undefined. In the context of the given curve, identifying critical points can help determine where the slope changes, which is essential for understanding the behavior of the curve. The points provided in the question are critical for evaluating the slope, as they represent specific locations on the curve where the slope needs to be calculated.
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04:50Critical Points
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In Exercises 17–28, find dy.
y = x√(1 − x²)
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In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
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In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.
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In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
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Free-Fall Applications
Free fall on Mars and Jupiter The equations for free fall at the surfaces of Mars and Jupiter (s in meters, t in seconds) are s = 1.86t² on Mars and s = 11.44t² on Jupiter. How long does it take a rock falling from rest to reach a velocity of 27.8 m/sec (about 100 km/h) on each planet?
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