2.1 The Tangent and Velocity Problems

Tangents

What is a tangent?

Tangent lines and Secant lines

Estimating slopes from discrete data:

Example:

1. A tank holds 1000 gallons of water, which drains from the bottom of the tank in halfan hour. The values in the table show the volume V of water remaining in the tank (ingallons) after t minutes.

t (min) 5 10 15 20 25 30V (gal) 694 444 250 111 28 0

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(a) If P is the point (15, 250) on the graph of V , find the slopes of the secant lines PQ whenQ is the point on the graph with t = 10 and t = 20.

(b) Estimate the slope of the tangent line at P by averaging the slopes of the two secantlines.

(c) Use a graph of the function to estimate the slope of the tangent line at P (representsrate at which water is flowing from the tank after 15 min).

Average Velocity vs. Instantaneous Velocity

Average velocity=

Instantaneous velocity at time t=

Example:

8. The displacement (in cm) of a particle moving back and forth along a straight line isgiven by the equation of motion s = 2 sinπt+ 3 cosπt, where t is measured in seconds.

(a) Find the average velocity during each period.

(i) [1,2] (ii) [1,1.1]

(iii) [1,1.01] (iv) [1,1.001]

(b) Estimate the instantaneous velocity of the particle when t = 1.

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2.2 The Limit of a Function

Definition The limit of f(x), as x approaches a, equals L written as limx→a

= L if f(x) can

be made arbitrarily close to L by taking x sufficiently close to a (on either side) but notequal to a.

Question What is the difference between f(a) = L and limx→a

f(x) = L?

Estimating using a table

Example: Create a table of values to estimate the limit.

19.

limx→1

x6 − 1x10 − 1

What about limx→0

sinπ

x?

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Conclusion: Although using a table is one way of estimating limits, this will not always givethe correct answer. So, we need to examine some other methods.

One-Sided Limits

limx→a

f(x) = L if and only if limx→a−

f(x) = L and limx→a+

f(x) = L

Example: For the given graph of g state the value for each of the quantities, if it exists. Ifit doesn’t exist, explain why.

5.

(a) limt→0−

g(t) (b) limt→0+

g(t) (c) limt→0

g(t)

(d) limt→2−

g(t) (e) limt→2+

g(t) (f) limt→2

g(t)

(g) g(2) (h)limt→4

g(t)

Summary

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2.3 Calculating Limits Using the Limit Laws

There are 11 limit laws listed on pages 108-110 in your book. Look these over to know whatyou can do with limits.

Examples of limit laws in action:

1. Given that limx→a

f(x) = −3, limx→a

g(x) = 0, and limx→a

h(x) = 8, find the limits that exist.

(a) limx→a

[f(x) + h(x)] (c) limx→a

3√h(x)

(f) limx→a

g(x)f(x)

(g) limx→a

f(x)g(x)

(h) limx→a

2f(x)h(x)− f(x)

Sometimes you can just plug in the numbers, and go. Other times algebraic manipulation(factoring, rationalizing the numerator, etc.) is needed.

Examples:

(a) limx→2

x2 − 4x− 2

(b) limx→0

x3 − 8x− 2

(c) limx→3

x3 − 8x− 2

(d) limx→2

x3 − 8x− 2

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(e) limx→1

√x+ 3− 2x− 1

When can and can’t you evaluate a limit just by plugging in the relevant value of x?

Examples:

34. limx→0+

(1x− 1|x|)

37. (a) (i) limx→−2+

[[x]] (ii) limx→−2

[[x]] (iii) limx→−2.4

[[x]]

(b) If n is an integer, evaluate (i) limx→n+

[[x]] and (ii) limx→n−

[[x]].

(c) For what values of a does limx→a

[[x]] exist?

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The Squeeze Theorem: If f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a)and lim

x→af(x) = lim

x→ah(a) = L then lim

x→ag(x)− L.

30. Prove that limx→0+

√xesin(π/x) = 0.

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2.4 Continuity

Definitions: A function f is continuous at a number a if limx→a

f(x) = f(a). If f is notcontinuous at a, we say f is discontinuous at a. Also, f is continuous on an intervalif f is continuous at every number in the interval.

This implies that three things need to happen for f to be continuous at a.

1.

2.

3.

What happens if 1 of these 3 properties is not satisfied?

Discontinuous=

Types of discontinuities

1. Removable 2. Infinite 3. Jump

Example: Find the discontinuities (if any). List the domain of each function.

a. f(x) =x− 1

x2 − 3x+ 2b. g(x) =

{x− 1 , x ≤ 1x+ 2 , x > 1

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NOTE: If f and g are continuous at a and c is a constant, then f + g, f − g, cf , fg, f/g(if g(a) 6= 0), and f ◦ g are also continuous at a.

This tells us the following functions are continuous on their domains:

Continuous from the left and right:

Intermediate Value Theorem

The Intermediate Value Theorem: Suppose that f is continuous on a closed interval[a, b] and let N be any number between f(a) and f(b), where f(a) 6= f(b). Then thereexists a number c in (a, b) such that f(c) = N .

IVT is used in a variety of ways!

1. Graphing Calculators

2. Proving that roots exist for a function

3. Tons of other applications

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Example: Show that there exists a number whose cube is one more than the number itself.

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2.5 Limits Involving Infinity

Definitions

1. limx→a

f(x) = ∞ if the values of f(x) can be made as large as we please by taking x

sufficiently close to a. Similar definition for −∞.

2. Let f be defined on (a,∞). Then limx→∞

f(x) = L if values of f(x) can be made as closeto L as we like by taking x sufficiently large. Similar for −∞.

Note: We also use limx→∞

f(x) = ∞ to indicate the function f is getting large as x becomes

large. Similar remarks can be made by replacing either (or both) ∞’s with −∞’s.

Some Important Examples

1. arctanx: 2. tanx:

3. ex: 4. lnx:

5.1x

:

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Asymptotes

What is a vertical asymptote? How do we find them?

What is a horizontal asymptote? How do we find them?

Calculating Limits

Examples:

Compute the limits of y =x+ 5x2 − 25

as x→ ±∞, and as x→ ±5. Graph the function.

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Compute limx→−∞

(x2 −√x4 + 2).

Graph y =2x3 − 16x3 − 27

after finding the vertical and horizontal asymptotes.

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2.6 Tangents, Velocities, and Other Rates of Change

Some of this should look familiar, we talked about it in .

Recall our definitions of tangent line and secant line.

“Locally Linear”

The tangent line to the curve y = f(x) at the point P (a, f(a)) is the line through P withslope

m = limx→a

f(x)− f(a)x− a

provided that limit exists.

Can also use an alternate definition:

m = limh→0

f(a+ h)− f(a)h

Example: 6. (a) Find the slope of the tangent line to the curve y = x3 at the point (−1,−1)

(b) Find an equation of the tangent line in part (a).

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Applications

For all rates of change (including velocity) the instantaneous rate of change is equal to the.

26. The number N of locations of a popular coffeehouse chain is given in the table.

Year 1998 1999 2000 2001 2002N 1886 2135 3501 4709 5886

(a) Find the average rate of growth from

(i) 2000 to 2002 (ii) 2000 to 2001 (iii) 1999 to 2000

(b) Estimate the instantaneous rate of growth in 2000 by taking the average of two averagerates of change. What are its units?

(c) Estimate the instantaneous rate of growth in 2000 by measuring the slope of the tangent.

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2.7 Derivatives

Definition of Derivative at a point a:

f ′(a) = limh→0

f(a+ h)− f(a)h

,

if this limit exists. This can also be written as:

f ′(a) = limx→a

f(x)− f(a)x− a

.

Example: Find the derivative of f(t) =√

1 + t at the number a.

Equation of tangent line

y − f(a) = f ′(a)(x− a)

Example: If f(t) =√

1 + t, find f ′(3) and use it to find an equation of the tangent line tof at the point (3, 2).

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Rate of Change

instantaneous rate of change=limx→0

4y4x

= limx2→x1

f(x2)− f(x1)x2 − x1

SO...the derivative f ′(a) = inst. rate of change of y = f(x) with respect to x when x = a.

When derivative is large, then .

When derivative is small, then .

Example:

25. A particle moves along a straight line w/ equation of motion s = f(t) = 100+50t−4.9t2,where s is measured in meters and t in seconds. Find the velocity and speed when t = 5.

Note 1: Velocity=

Note 2: Speed=

30. The quantity (in pounds) of a coffee that is sold at a price of p dollars per pound isQ = f(p).

(a) What is the meaning of the derivative f ′(8)? What are its units?

(b)Is f ′(8) positive or negative? Explain.

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2.8 The Derivative as Function

Creating f ′ from f :

4.

Let’s look at some other notations for the derivative:

Definition: A function f is differentiable at a if f ′(a) exists. The function f is differ-entiable on an open interval (a, b) if it is differentiable at every number in the interval.

Theorem If f is differentiable at a, then f is continuous at a.

When is a function not differentiable?

1. 2. 3.

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The converse of the above theorem is not true. Can you give me an example?

Higher Derivatives

We can also take the second derivative, third derivative, etc. of a function. In order to take ahigher derivative, .

Notation:

Position Functions and Higher Derivatives:

39. The graph shows three functions. Determine which one is the position function of acar, the velocity function of the car, and the acceleration.

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2.9 What does f ′ say about f?

Increasing/Decreasing

If f ′(x) > 0, then f is .

If f ′(x) < 0, then f is .

Example:

11.

(a) On what intervals is f increasing or decreasing?

(b) At what values of x does f have a local maximum or minimum?

Concavity

If f ′′(x) > 0, then f is .

If f ′′(x) < 0, then f is .

Example:

11.

(c) On what intervals is f concave upward or downward?

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An inflection point is a point on the graph where the second derivative changes sign, i.e.where the concavity changes.

(d) State the x-coordinates of the point(s) of inflection.

(e) Assuming f(0) = 0, sketch a graph of f .

Example:

Given the following graph of f , sketch f ′ and f ′′.

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Example:

19. Sketch the graph of a function that satisfies: f ′(x) > 0 if |x| < 2, f ′(x) < 0 if |x| > 2,f ′(−2) = 0, lim

x→2|f ′(x)| =∞, and f ′′(x) > 0 if x 6= 2.

An antiderivative of f is a function F such that F ′ = f .

If F is an antiderivative of f , so is .

Example:

28. The graph of the velocity of a car is shown. Sketch the graph of the position function.

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