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MAT 1234 Calculus I Section 1.4 The Tangent and Velocity Problems http://myhome.spu.edu/lauw

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WebAssign Homework 1.4 (Only 2 Problems!) Have a nice weekend!

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Two Worlds and Two Problems ?

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What do we care? How fast “things” are going The velocity of a particle The “speed” of formation of chemicals The rate of change of a population

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What is “Rate of Change”? Rate of ChangeContext 60 miles/hour at t=40s 30 ml/s at t=5s -30ml/s at t=5s -$5/min at t=8:05am

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What is “Rate of Change”? We are going to look at how to understand and how to find the “ rate of change ” in terms of functions.

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The Problems The Tangent Problem The Velocity Problem

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Example 1 The Tangent Problem Slope=?

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Example 1 The Tangent Problem We are going to use an “limiting” process to “find” the slope of the tangent line at x=1. Slope=?

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Example 1 The Tangent Problem First we compute the slope of the secant line between x=1 and x=3. Slope=?

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Example 1 The Tangent Problem Then we compute the slope of the secant line between x=1 and x=2. Slope=?

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Example 1 The Tangent Problem As the point on the right hand side of x=1 getting closer and closer to x=1, the slope of the secant line is getting closer and closer to the slope of the tangent line at x=1. Slope=?

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Example 1 The Tangent Problem First we compute the slope of the secant line between x=1 and x=3. Slope=?

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Example 1 The Tangent Problem First we compute the slope of the secant line between x=1 and x=3. Slope=?

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Example 1 The Tangent Problem Let us record the results in a table. hslope 22 1 0.1 0.01

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Example 1 The Tangent Problem We see from the table that the slope of the tangent line at x=1 should be _________.

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Limit Notations When h is approaching 0, is approaching ___. We say as h 0, Or,

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Definition For the graph of, the slope of the tangent line at is if it exists.

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Two Worlds and Two Problems

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Example 2 The Velocity Problem y = distance dropped (ft) t = time (s) Displacement Function (Positive Downward) Find the velocity of the ball at t=2.

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Example 2 The Velocity Problem Again, we are going to use the same “limiting” process. Find the average velocity of the ball from t=2 to t=2+h by the formula

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Example 2 The Velocity Problem thAverage Velocity (ft/s) 2 to 31 2 to 2.10.1 2 to 2.010.01 2 to 2.0010.001

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Example 2 The Velocity Problem We see from the table that velocity of the ball at t=2 should be _________ft/s.

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Limit Notations When h is approaching 0, is approaching____. We say as h 0, Or,

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Definition For the displacement function, the instantaneous velocity at is if it exists.

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Two Worlds and Two Problems

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Review and Preview Example 1 and 2 show that in order to solve the tangent and velocity problems we must be able to find limits. In the next few sections, we will study the methods of computing limits without guessing from tables.

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Tangents, Velocities, and Other Rates of Change Definition The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope.

Tangents, Velocities, and Other Rates of Change Definition The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope.

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