Domanda del libro di testo
60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>
b. Graph the tangent lines on the given graph.
x+y³−y=1; x=1
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.8.14b
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
x = e^y; (2, ln 2)
Guida verificata passo dopo passo1
Start by recognizing that the equation given is in implicit form: \( x = e^y \). This means that \( y \) is not isolated on one side of the equation.
To find the derivative \( \frac{dy}{dx} \), apply implicit differentiation to both sides of the equation with respect to \( x \). Differentiate \( x \) to get 1, and differentiate \( e^y \) using the chain rule to get \( e^y \cdot \frac{dy}{dx} \).
Set up the equation from the differentiation: \( 1 = e^y \cdot \frac{dy}{dx} \).
Solve for \( \frac{dy}{dx} \) by isolating it on one side of the equation: \( \frac{dy}{dx} = \frac{1}{e^y} \).
Substitute the given point \((2, \ln 2)\) into the expression for \( \frac{dy}{dx} \). Since \( y = \ln 2 \), calculate \( e^{\ln 2} \) which simplifies to 2, and find the slope at this point.
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Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for curves defined by equations that cannot be easily rearranged.
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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 represents the rate of change of the function at that point, which is mathematically defined as the derivative of the function. For a curve defined implicitly, the slope can be found by evaluating the derivative obtained through implicit differentiation. The slope is often denoted as 'dy/dx' and indicates how steep the curve is at the specified coordinates.
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11:41Summary of Curve Sketching
Exponential and Natural Logarithm Functions
Exponential functions, such as e^y, and natural logarithm functions, like ln(x), are fundamental in calculus. The function e^y is the inverse of ln(y), and they exhibit unique properties, such as the derivative of e^y being e^y dy/dx. Understanding these functions is crucial for evaluating expressions and derivatives involving exponential growth or decay, especially when working with implicit relationships.
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05:18Derivative of the Natural Logarithmic Function
Pratica correlata
Domanda del libro di testo
13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
(x+y)^2/3=y; (4, 4)
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Domanda del libro di testo
{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.
b. Graph the volume function and verify that the tank is empty when t=1000 min.
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Domanda del libro di testo
Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.
b. When does the stone reach its highest point?
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Domanda del libro di testo
Product Rule for three functions Assume f, g, and h are differentiable at x.
b. Use the formula in (a) to find d/dx(e^x(x−1)(x+3))
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Domanda del libro di testo
Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>
b. What is the rate of change of θ with respect to the bird’s height when it is 60 ft above the ground?
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