Textbook Question
Finding Functions from Derivatives
In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.
r'(t) = sec t tan t − 1, P(0, 0)
149
views
4. Applications of Derivatives
Differentials
2:20 minutesProblem 54a
Textbook Question
When the length L of a clock pendulum is held constant by controlling its temperature, the pendulum’s period T depends on the acceleration of gravity g. The period will therefore vary slightly as the clock is moved from place to place on Earth’s surface, depending on the change in g. By keeping track of ΔT, we can estimate the variation in g from the equation T = 2π(L/g)¹/² that relates T, g, and L.
a. With L held constant and g as the independent variable, calculate dT and use it to answer parts (b) and (c).
Verified step by step guidance1
Start by identifying the given equation for the pendulum's period: T = 2π(L/g)¹/². Here, L is constant, and g is the variable.
To find dT, differentiate T with respect to g using implicit differentiation. Since L is constant, treat it as a constant during differentiation.
Apply the chain rule to differentiate T = 2π(L/g)¹/² with respect to g. The derivative of (L/g)¹/² with respect to g is -1/2 * (L/g)⁻³/² * (L/g²).
Multiply the derivative by the constant factor 2π to get dT/dg = -π(L/g)⁻³/² * (L/g²).
Express dT in terms of dg: dT = -π(L/g)⁻³/² * (L/g²) * dg. This equation can be used to estimate the variation in g based on changes in T.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differentiation
Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of a function with respect to a variable. In this context, we need to differentiate the period T with respect to the acceleration due to gravity g, treating L as a constant, to find dT, the change in the period.
Recommended video:
05:53
Finding Differentials
Chain Rule
The chain rule is a technique used in calculus to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, the chain rule helps in differentiating T = 2π(L/g)¹/² with respect to g, as it involves a composition of functions.
Recommended video:
05:02Intro to the Chain Rule
Partial Derivatives
Partial derivatives are used to find the derivative of a function with respect to one variable while keeping other variables constant. In this problem, since L is held constant, we treat T as a function of g alone and find the partial derivative of T with respect to g. This helps in understanding how changes in g affect the period T of the pendulum.
Recommended video:
05:44Derivatives
Related Videos
Related Practice