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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.6.59a
Chapter 3, Problem 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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Key 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 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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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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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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Related Practice
Textbook Question

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4x³ =y²(4−x); x=2 (cissoid of Diocles)

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Textbook Question

9–61. Evaluate and simplify y'.


y = tan^−1 √t²−1

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a. At which points is the slope of the curve negative?

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Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = -7x; P(-1,7)

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Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = 2/√x; P(4,1)

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