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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.8.58b
Chapter 3, Problem 3.8.58b

58–59. Carry out the following steps.
b. Find the slope of the curve at the given point.
xy^5/2+x^3/2y=12; (4, 1)

Verified step by step guidance
1
First, understand that the problem requires finding the slope of the curve at a specific point, which involves calculating the derivative of the given equation implicitly.
The given equation is \( xy^{\frac{5}{2}} + x^{\frac{3}{2}}y = 12 \). To find the derivative, we will use implicit differentiation since the equation involves both x and y.
Differentiate both sides of the equation with respect to x. Remember to apply the product rule where necessary. For example, when differentiating \( xy^{\frac{5}{2}} \), treat it as a product of x and \( y^{\frac{5}{2}} \).
After differentiating, you will have an expression involving \( \frac{dy}{dx} \). Solve this expression for \( \frac{dy}{dx} \) to find the derivative, which represents the slope of the curve.
Finally, substitute the given point (4, 1) into the derivative expression to find the slope at that specific point.

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Key Concepts

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

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, we differentiate both sides of the equation with respect to x, treating y as a function of x. This allows us to find dy/dx, which represents the slope of the curve at any point.
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Finding The Implicit Derivative

Slope of a Curve

The slope of a curve at a given point is defined as the rate of change of the y-coordinate with respect to the x-coordinate at that point. Mathematically, it is represented by the derivative of the function at that point. For implicit functions, the slope can be found by evaluating dy/dx after applying implicit differentiation.
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Summary of Curve Sketching

Evaluating Derivatives at a Point

Once the derivative (dy/dx) is found using implicit differentiation, the next step is to evaluate it at a specific point, in this case, (4, 1). This involves substituting the x and y values into the derivative expression to calculate the exact slope of the curve at that point, providing insight into the curve's behavior.
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Critical Points
Related Practice
Textbook Question

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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Textbook Question

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.

x³+y³=2xy; (1, 1)

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Textbook Question

Consider the following cost functions.

b. Determine the average cost and the marginal cost when x=a.

C(x) = − 0.01x²+40x+100, 0≤x≤1500, a=1000

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Textbook Question

Deriving trigonometric identities

b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

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21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(t) = 1/√t; a=9, 1/4

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{Use of Tech} Computing limits with angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.

b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.

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