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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 3 - Derivatives문제 16
3장, 문제 16

Slopes and Tangent Lines


In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.


y = (x + 3)/(1 – x), x = −2

검증된 단계별 안내
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Step 1: Identify the function to differentiate. The function given is \( y = \frac{x + 3}{1 - x} \).
Step 2: Apply the quotient rule for differentiation. The quotient rule states that if you have a function \( y = \frac{u}{v} \), then its derivative \( y' \) is given by \( y' = \frac{u'v - uv'}{v^2} \), where \( u = x + 3 \) and \( v = 1 - x \).
Step 3: Differentiate the numerator and the denominator separately. For \( u = x + 3 \), the derivative \( u' = 1 \). For \( v = 1 - x \), the derivative \( v' = -1 \).
Step 4: Substitute \( u, u', v, \) and \( v' \) into the quotient rule formula: \( y' = \frac{(1)(1 - x) - (x + 3)(-1)}{(1 - x)^2} \). Simplify the expression to find \( y' \).
Step 5: Evaluate the derivative at \( x = -2 \) to find the slope of the tangent line. Substitute \( x = -2 \) into the simplified expression for \( y' \) and calculate the slope.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's value changes with respect to changes in its input. For a function y = f(x), the derivative, denoted as f'(x) or dy/dx, provides the slope of the tangent line to the curve at any point x. Understanding differentiation is crucial for solving problems involving rates of change and slopes of curves.
추천 영상:
가이드 코스
05:53
Finding Differentials

Quotient Rule

The Quotient Rule is a method for differentiating functions that are expressed as a quotient of two other functions, u(x) and v(x). It states that the derivative of y = u(x)/v(x) is given by (v(x)u'(x) - u(x)v'(x))/(v(x))^2. This rule is essential when dealing with rational functions, like y = (x + 3)/(1 - x), to find their derivatives accurately.
추천 영상:
06:43
The Quotient Rule

Slope of the Tangent Line

The slope of the tangent line to a curve at a given point is the value of the derivative of the function at that point. It represents the instantaneous rate of change of the function and is crucial for understanding the behavior of the function at specific values of the independent variable. For the function y = (x + 3)/(1 - x) at x = -2, calculating the derivative and evaluating it at x = -2 gives the slope of the tangent line.
추천 영상:
가이드 코스
05:13
Slopes of Tangent Lines
관련 실천
교과서 질문

Understanding Motion from Graphs


Launching a Rocket When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive charge pops out a parachute shortly after the rocket starts down. The parachute slows the rocket to keep it from breaking when it lands.


The figure here shows velocity data from the flight of the model rocket. Use the data to answer the following.


c. When did the rocket reach its highest point? What was its velocity then?


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교과서 질문

Understanding Motion from Graphs


Launching a Rocket When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive charge pops out a parachute shortly after the rocket starts down. The parachute slows the rocket to keep it from breaking when it lands.


The figure here shows velocity data from the flight of the model rocket. Use the data to answer the following.


b. For how many seconds did the engine burn?


205
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교과서 질문

Understanding Motion from Graphs


Launching a Rocket When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive charge pops out a parachute shortly after the rocket starts down. The parachute slows the rocket to keep it from breaking when it lands.


The figure here shows velocity data from the flight of the model rocket. Use the data to answer the following.


a. How fast was the rocket climbing when the engine stopped?


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교과서 질문

Slopes and Tangent Lines


In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.


f(x) = x + 9/x, x = −3

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교과서 질문

Finding Derivative Functions and Values 


Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.


p(θ) = √3θ; p′(1), p′(3), p′(2/3)

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교과서 질문

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.


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Find the derivatives with respect to x of the following combinations at the given value of x.


h. √(f²(x) + g²(x)), x = 2

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