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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.8.48b
Capitolo 3, Problema 3.8.48b

45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
x⁴-x²y+y⁴=1; (−1, 1)

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First, understand that the problem requires finding the equation of the tangent line to the curve defined by the equation x⁴ - x²y + y⁴ = 1 at the point (-1, 1). This involves using implicit differentiation to find the derivative dy/dx.
Differentiate both sides of the equation x⁴ - x²y + y⁴ = 1 with respect to x. Remember to apply the product rule to the term x²y and the chain rule to y⁴. The derivative of x⁴ is 4x³, and the derivative of y⁴ with respect to x is 4y³(dy/dx).
For the term x²y, apply the product rule: the derivative is 2xy + x²(dy/dx). Combine all these derivatives to form the equation: 4x³ - (2xy + x²(dy/dx)) + 4y³(dy/dx) = 0.
Solve the resulting equation for dy/dx, which represents the slope of the tangent line at any point (x, y) on the curve. Substitute the given point (-1, 1) into the equation to find the specific slope at that point.
Once you have the slope, use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is the given point (-1, 1), to write the equation of the tangent line.

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Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function defined implicitly by an equation involving both x and y. In this case, the equation x⁴ - x²y + y⁴ = 1 requires us to differentiate both sides with respect to x, treating y as a function of x. This allows us to find dy/dx, which is essential for determining the slope of the tangent line at a specific point.
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Percorso guidato
05:14
Finding The Implicit Derivative

Tangent Line Equation

The equation of a tangent line at a given point on a curve can be expressed using the point-slope form: y - y₀ = m(x - x₀), where (x₀, y₀) is the point of tangency and m is the slope of the tangent line. Once the slope is calculated using implicit differentiation, this formula can be applied to find the specific equation of the tangent line at the point (-1, 1) on the curve.
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Percorso guidato
05:14
Equations of Tangent Lines

Slope of the Tangent Line

The slope of the tangent line at a point on a curve represents the instantaneous rate of change of the function at that point. It is calculated as the derivative of the function evaluated at the specific x-coordinate. In this problem, finding the slope at the point (-1, 1) is crucial for constructing the tangent line equation, as it directly influences the line's steepness and direction.
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Slopes of Tangent Lines
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Domanda del libro di testo

97–100. Logistic growth Scientists often use the logistic growth function P(t) = P₀K / P₀+(K−P₀)e^−r₀t to model population growth, where P₀ is the initial population at time t=0, K is the carrying capacity, and r₀ is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. <IMAGE>


{Use of Tech} Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is P(t) = 400,000 / 50+7950e^−0.5t, where t is measured in years.


b. How long does it take for the population to reach 5000 fish? How long does it take for the population to reach 90% of the carrying capacity?

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Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

b. (f^-1)'(6)

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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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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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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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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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