Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 8x; a = −3
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 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?
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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06:11Fundamental Theorem of Calculus Part 1
Related Practice
Textbook Question
Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.
f(x) = -3x2 - 5x + 1; P(1,-7)
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Textbook Question
{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>
a. What is the rate of change of the angle of elevation dθ/dx when the plane is x=500 m past the observer?
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Textbook Question
Derivatives using tables Let and . Use the table to compute the following derivatives.
<IMAGE>
a.
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Textbook Question
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
x = e^y; (2, ln 2)
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Textbook Question
{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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