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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 3.6.59a
Capitolo 3, Problema 3.6.59a

A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e-t cos t), for t ≥ 0.
Determine her velocity at t = 1 and t = 3. 

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1
To find the velocity of the woman at a given time, we need to determine the derivative of her height function y(t) with respect to time t. The derivative, y'(t), represents the velocity.
The height function is given as y(t) = 15(1 + e^(-t) * cos(t)). We will apply the product rule and chain rule to differentiate this function.
First, identify the components of the function: u(t) = e^(-t) and v(t) = cos(t). The product rule states that the derivative of u(t) * v(t) is u'(t) * v(t) + u(t) * v'(t).
Calculate the derivatives: u'(t) = -e^(-t) (using the chain rule for e^(-t)) and v'(t) = -sin(t) (derivative of cos(t)).
Substitute these derivatives into the product rule formula: y'(t) = 15 * [(-e^(-t) * cos(t)) + (e^(-t) * (-sin(t)))]. Evaluate y'(t) at t = 1 and t = 3 to find the velocity at these times.

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Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable, which in this case is time. To determine the velocity of the woman at specific times, we need to differentiate her height function y(t) with respect to t, yielding the velocity function v(t).
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Finding Differentials

Exponential Functions

Exponential functions are mathematical functions of the form f(t) = a * e^(bt), where e is Euler's number. In the given height function y(t), the term e^(-t) indicates that the height changes exponentially over time. Understanding how exponential decay affects the height is crucial for accurately calculating the velocity at different time points.
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Exponential Functions

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In the height function y(t), the cosine term introduces oscillatory behavior to the height over time. Recognizing how the cosine function influences the overall height is important for evaluating the velocity at specific moments during the jump.
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Introduction to Trigonometric Functions
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