Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 ( x- 1)^sinπx
4. Applications of Derivatives
Differentials
2:15 minutesProblem 4.6.65
Textbook Question
Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.
f(x) = 2 - a cos x, a constant
Verified step by step guidance1
Identify the function given: \( f(x) = 2 - a \cos x \), where \( a \) is a constant.
To find the differential \( dy \), we first need to compute the derivative of \( f(x) \) with respect to \( x \).
Differentiate \( f(x) \) using the derivative of cosine: \( \frac{d}{dx}[\cos x] = -\sin x \). Thus, \( f'(x) = 0 - a(-\sin x) = a \sin x \).
Express the differential \( dy \) in terms of \( dx \) using the derivative: \( dy = f'(x)dx = a \sin x \, dx \).
The relationship between a small change in \( x \) and the corresponding change in \( y \) is given by \( dy = a \sin x \, dx \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentials
Differentials represent the infinitesimally small changes in variables. In calculus, if y is a function of x, the differential dy is defined as dy = f'(x)dx, where f'(x) is the derivative of the function at a point x, and dx is a small change in x. This relationship helps in approximating how y changes in response to small changes in x.
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Derivatives
The derivative of a function measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In the context of the given function, f'(x) will provide the slope of the tangent line at any point, indicating how y changes with respect to x.
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Chain Rule
The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This concept is essential when dealing with functions that involve constants or other functions, as it allows for the correct computation of derivatives in complex scenarios.
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