Domanda del libro di testo
If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at x=1, 2, and 3.
306
views
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 3.11.17
A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?
Guida verificata passo dopo passo1
First, recall the formula for the area of a circle: A = πr², where A is the area and r is the radius.
To find the rate of change of the area with respect to time, we need to differentiate the area formula with respect to time. This involves using the chain rule since the radius is changing with time.
Apply the chain rule: dA/dt = dA/dr * dr/dt. Here, dA/dr is the derivative of the area with respect to the radius, and dr/dt is the rate at which the radius is changing.
Calculate dA/dr: Differentiate A = πr² with respect to r to get dA/dr = 2πr.
Substitute the values: Use r = 10 ft and dr/dt = -2 ft/min (since the radius is decreasing) into the equation dA/dt = 2πr * dr/dt to find the rate of change of the area at the instant the radius is 10 ft.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
6mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Related Rates
Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how the area of the circle changes as the radius decreases. This requires applying the chain rule from calculus to relate the rates of change of the radius and the area.
Video consigliato:
Percorso guidato
04:16
Intro To Related Rates
Area of a Circle
The area of a circle is calculated using the formula A = πr², where A is the area and r is the radius. Understanding this formula is crucial because we need to differentiate it with respect to time to find how the area changes as the radius changes over time.
Video consigliato:
Percorso guidato
05:59Estimating the Area Under a Curve with Right Endpoints & Midpoint
Differentiation
Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. In this context, we will differentiate the area formula with respect to time to find the rate of change of the area as the radius changes, applying the chain rule to connect the rates of change.
Video consigliato:
Percorso guidato
05:53Finding Differentials
Pratica correlata
Domanda del libro di testo
Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (tan 7x) / (sin x)
358
views
Domanda del libro di testo
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
y = (cos x) In cos²x
206
views
Domanda del libro di testo
13–40. Evaluate the derivative of the following functions.
f(x) = 1/tan^−1(x²+4)
229
views
Domanda del libro di testo
Find the slope of the curve x²+y³=2 at each point where y=1 (see figure). <IMAGE>
262
views
Domanda del libro di testo
Find the derivative of the following functions.
y = x² (1 - In x²)
215
views