Domanda del libro di testo
Given that f(1)=2 and f′(1)=2 , find the slope of the curve y=xf(x) at the point (1, 2).
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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 3.11.56
Two boats leave a port at the same time, one traveling west at 20 mi/hr and the other traveling southwest ( 45° south of west) at 15 mi/hr. After 30 minutes, how far apart are the boats and at what rate is the distance between them changing? (Hint: Use the Law of Cosines.)
Guida verificata passo dopo passo1
First, determine the distance each boat has traveled after 30 minutes. Since 30 minutes is 0.5 hours, the westbound boat travels 20 mi/hr * 0.5 hr = 10 miles, and the southwest-bound boat travels 15 mi/hr * 0.5 hr = 7.5 miles.
Next, set up a triangle where the two boats are at the ends of two sides, and the angle between their paths is 45°. The westbound boat's path is one side, and the southwest-bound boat's path is the other side.
Use the Law of Cosines to find the distance between the two boats. The Law of Cosines states: c^2 = a^2 + b^2 - 2ab * cos(C), where a and b are the sides of the triangle, and C is the angle between them. Here, a = 10 miles, b = 7.5 miles, and C = 45°.
Substitute the known values into the Law of Cosines: c^2 = 10^2 + 7.5^2 - 2 * 10 * 7.5 * cos(45°). Calculate the cosine of 45°, which is √2/2, and substitute it into the equation.
Finally, to find the rate at which the distance between the boats is changing, differentiate the Law of Cosines equation with respect to time. This involves using implicit differentiation and considering the rates of change of the distances traveled by each boat.
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Law of Cosines
The Law of Cosines is a formula used in geometry to find the lengths of sides in a triangle when two sides and the included angle are known. It states that for any triangle with sides a, b, and c, and angle C opposite side c, the relationship is c² = a² + b² - 2ab * cos(C). This law is particularly useful in this problem to determine the distance between the two boats after they have traveled for a certain time.
Video consigliato:
4:49
Inverse Cosine
Relative Velocity
Relative velocity refers to the velocity of one object as observed from another object. In this scenario, understanding how the velocities of the two boats interact is crucial for determining how quickly the distance between them is changing. By analyzing their velocities in terms of components, we can calculate the rate at which the distance between the boats increases.
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Percorso guidato
06:29Derivatives Applied To Velocity
Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, relate the angles and sides of triangles. In this problem, the angle at which the second boat travels (southwest) is essential for breaking down its velocity into components. These functions will help in calculating the positions of the boats and ultimately in applying the Law of Cosines to find the distance between them.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
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