Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 24f

Chapter 3, Problem 24f

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 19.6 m/s from a height of 24.5 m above the ground. The height (in meters) of the stone above the ground t seconds after it is thrown is s(t) = -4.9t² + 19.6t + 24.5.
On what intervals is the speed increasing?

Verified step by step guidance

Step 1: Understand that the speed of the stone is increasing when the magnitude of its velocity is increasing. The velocity function v(t) is the derivative of the height function s(t).

Step 2: Differentiate the height function s(t) = -4.9t^2 + 19.6t + 24.5 to find the velocity function v(t). This gives v(t) = s'(t) = -9.8t + 19.6.

Step 3: The speed is increasing when the derivative of the velocity, which is the acceleration, has the same sign as the velocity. Calculate the acceleration by differentiating the velocity function: a(t) = v'(t) = -9.8.

Step 4: Since the acceleration a(t) = -9.8 is constant and negative, the velocity is decreasing when it is positive and increasing when it is negative. Find when v(t) = 0 to determine when the velocity changes sign.

Step 5: Solve the equation v(t) = -9.8t + 19.6 = 0 to find the critical point where the velocity changes sign. This will help determine the intervals where the speed is increasing.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Velocity and Speed

Velocity is a vector quantity that refers to the rate of change of position with respect to time, including direction. Speed, on the other hand, is the magnitude of velocity and does not consider direction. In this context, understanding how velocity changes over time is crucial for determining when the speed of the stone is increasing.

Acceleration

Acceleration is the rate of change of velocity with respect to time. In the case of the stone thrown upward, the acceleration due to gravity is acting downward at approximately -9.8 m/s². Analyzing the acceleration helps determine whether the speed of the stone is increasing or decreasing, particularly when the velocity changes sign.

Critical Points and Intervals

Critical points occur where the derivative of a function is zero or undefined, indicating potential maxima, minima, or points of inflection. To find intervals where speed is increasing, one must analyze the derivative of the velocity function, identifying where it is positive. This involves determining the intervals on the time axis where the speed of the stone is increasing based on the sign of the derivative.

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Critical Points

Related Practice

Textbook Question

Suppose a baseball is thrown vertically upward from the ground with an initial velocity of v₀ ft/s. The approximate height of the ball (in feet) above the ground after t seconds is given by s(t) = -16t² + v₀t.

With what velocity does the ball strike the ground?

Textbook Question

Once Kate’s kite reaches a height of 50 ft (above her hands), it rises no higher but drifts due east in a wind blowing 5 ft/s. How fast is the string running through Kate’s hands at the moment when she has released 120 ft of string?

Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t² + 32t + 48.

On what intervals is the speed increasing?

Textbook Question

With what velocity does the stone strike the ground?

Textbook Question

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 64 ft/s from a height of 32 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t² + 64t + 32.

On what intervals is the speed increasing?

Textbook Question

Evaluate the derivative of the following functions.

f(x) = x² + 2x³ cot^-1 x - ln (1 + x²)