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A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume $V$ of water remaining in the tank (in gallons) after $t$ minutes. $$\begin{array}{|c|c|c|c|c|c|c|}\hline t(\mathrm{~min}) & 5 & 10 & 15 & 20 & 25 & 30 \\\hline V(\mathrm{gal}) & 694 & 444 & 250 & 111 & 28 & 0 \\\hline\end{array}$$(a) If $P$ is the point (15,250) on the graph of $V$, find the slopes of the secant lines $P Q$ when $Q$ is the point on the graph with $t=5,10,20,25,$ and 30(b) Estimate the slope of the tangent line at $P$ by averaging the slopes of two secant lines.(c) Use a graph of the function to estimate the slope of the tangent line at $P$. (This slope represents the rate at which the water is flowing from the tank after 15 minutes.)

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 1

The Tangent and Velocity Problems

Limits

Derivatives

Wilson S.

December 8, 2020

Prince O.

March 17, 2019

thanks

Campbell University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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A tank holds 1000 gallons …

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A tank holds 1000 liters o…

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A tank holds 1000 gall…

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The rate at which water is…

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If a cylindrical tank hold…

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