Domanda del libro di testo
45–50. Tangent lines Carry out the following steps. <IMAGE>
a. Verify that the given point lies on the curve.
x³+y³=2xy; (1, 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 3.11.14a
Shrinking isosceles triangle The hypotenuse of an isosceles right triangle decreases in length at a rate of 4 m/s.
a. At what rate is the area of the triangle changing when the legs are 5 m long?
Guida verificata passo dopo passo1
Identify the relationship between the hypotenuse and the legs of the isosceles right triangle. In an isosceles right triangle, the hypotenuse \( c \) is related to the legs \( a \) by the equation \( c = a\sqrt{2} \).
Differentiate the equation \( c = a\sqrt{2} \) with respect to time \( t \) to find the relationship between the rates of change of the hypotenuse and the legs. This gives \( \frac{dc}{dt} = \sqrt{2} \frac{da}{dt} \).
Substitute the given rate of change of the hypotenuse \( \frac{dc}{dt} = -4 \) m/s into the differentiated equation to solve for \( \frac{da}{dt} \), the rate of change of the legs.
Use the formula for the area \( A \) of an isosceles right triangle, \( A = \frac{1}{2}a^2 \), and differentiate it with respect to time \( t \) to find \( \frac{dA}{dt} \). This gives \( \frac{dA}{dt} = a \frac{da}{dt} \).
Substitute \( a = 5 \) m and the previously calculated \( \frac{da}{dt} \) into the differentiated area formula to find the rate at which the area is changing, \( \frac{dA}{dt} \).
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Related Rates
Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to relate the rate of change of the hypotenuse to the rate of change of the area of the triangle. This requires the use of derivatives to express how changes in one variable affect another over time.
Video consigliato:
Percorso guidato
04:16
Intro To Related Rates
Area of a Triangle
The area of a triangle can be calculated using the formula A = (1/2) * base * height. For an isosceles right triangle, the legs are equal, and both serve as the base and height. Understanding how to express the area in terms of the leg length is crucial for determining how the area changes as the triangle shrinks.
Video consigliato:
Percorso guidato
07:59Estimating the Area Under a Curve Using Left Endpoints
Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is essential for relating the lengths of the legs of the triangle to the hypotenuse, allowing us to express the leg lengths in terms of the hypotenuse as it changes over time.
Video consigliato:
Percorso guidato
06:11Fundamental Theorem of Calculus Part 1
Pratica correlata
Domanda del libro di testo
7–14. Find the derivative the following ways:
a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.
f(w) = w³ -w / w
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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>
a. Express the angle of elevation θ from the biologist to the falcon as a function of the height h of the bird above the ground. (Hint: The vertical distance between the top of the cliff and the falcon is 80−h.)
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Domanda del libro di testo
Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>
a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).
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Domanda del libro di testo
{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by
f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>
Let f (x) = √x.
a. Find the exact value of f' (4).
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Domanda del libro di testo
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
sin y = 5x⁴−5; (1, π)
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