AP CALCULUS BC – PACKET FOR UNIT 1 LIMITS AND CONTINUITY
Tangent and Secant Lines
1. From geometry:
a. Explain what it means for a line to be tangent to a circle.
b. What is a secant line of a circle?
In calculus, we study the properties of secant lines and tangent lines to a curve. A secant line
connects any two points on a curve. A tangent line skims a curve at a single point called a
point of tangency.
Secant line
Tangent line
1. Let f ( x=
)
1+ x .
a. Graph f ( x ) , and sketch the secant line from x = 2
to x = 4 .
b. Find the slopes of the secant lines on each
interval. See the bottom of the page for some
calculator directions.
i.
x = 2 to x = 4
ii.
x = 2.5 to x = 3.5
iii. x = 2.8 to x = 3.2
Finding Slope on the Calculator:
• Type the function in Y=.
• Go to the home screen by pressing 2ND MODE.
• Type the fraction bar by pressing ALPHA Y= ENTER. If this doesn’t give you a fraction
bar, then you don’t have the newest operating system; ask Ms. Lee for help.
• Type Y1 by pressing ALPHA TRACE ENTER.
Y1( 4 ) − Y1( 2 )
• Complete the expression like this:
4−2
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AP CALCULUS BC – PACKET FOR UNIT 1 LIMITS AND CONTINUITY
1. (continued)
c. Use your answers from part (b) to estimate the slope of the tangent line at x = 3 .
d. Find an equation of the tangent line to the curve at x = 3 . Leave your answer in pointslope form.
2. Let f=
( x ) 0.1x3 − 3x .
a. Sketch the graph of f.
b. Where is f increasing? Decreasing?
c. Where is the slope of the tangent line
positive? Negative? Zero? How are these
answers related to your answers in (b)?
3. Jason was stuck in stop-and-go traffic on I-95. He was so bored that he recorded the
distance he’d traveled at certain intervals, as shown in the table below:
Time, t (hours
elapsed)
Cumulative
Distance (miles)
0
1
2
3
4
5
0
20
35
45
52
57
a. Average velocity is
∆distance
. Calculate Jason’s average velocity from t = 2 to t = 4 .
∆time
b. How could you use the table to estimate Jason’s instantaneous velocity at t = 2 ?
c. How are average and instantaneous velocity related to the secant and tangent lines
of the distance vs. time graph?
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AP CALCULUS BC – PACKET FOR UNIT 1 LIMITS AND CONTINUITY
4. Suppose you try to use the technique you described in #3b to find the slope of the
tangent line at x = 0 for each of these functions. Use graphs and tables on your calculator
to determine what will happen in each case.
a. f ( x ) = 3 x
1
c. h ( x ) = sin  
x
b. g ( x ) = x
5. Take a textbook from the cabinet, and complete Classwork 2.1.
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AP CALCULUS BC – PACKET FOR UNIT 1 LIMITS AND CONTINUITY
CALCULATOR ACTIVITY –TANGENT LINES AND LIMITS
The method you learned in section 2.1 for estimating the slope of a tangent line requires you
to find the limit of the slopes of the secant lines. In other words, you determined the value
these slopes approached. In order to use limits to find slopes of tangent lines more
effectively, we’ll explore limits further…
sin x
. At first glance, the function appears to exist everywhere,
x
but you know by looking at the equation that it does not exist at 0. However, the graph
appears to approach the same point from either side of 0. You can probably guess what
value f approaches as x nears 0, but let’s look at how to use the calculator to find that:
1. Look at the graph of f ( x ) =
a. Press TRACE. Type a number close to 0, and press ENTER. Note the value of y, and
repeat for a few values successively closer to 0. What value does y seem to
approach?
b. If you used positive numbers in part (a), try the same thing with negative x-values,
gradually drawing closer to 0. You should get the same answer as in part (a).
sin x
sin x
, read as “the limit, as x approaches 0, of
.”
x →0
x
x
The notation for this limit is: lim
1 1
+
e −1
2
x.
2. Use your calculator to estimate the value of lim
and lim
x→−2 2 + x
x →0
x
2x
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AP CALCULUS BC – PACKET FOR UNIT 1 LIMITS AND CONTINUITY
PRACTICE PROBLEMS FOR UNIT 1
The problems below can be found on the Calculus websites for UC Davis, Oklahoma University, and
Dartmouth College. Unless a problem says to “use a calculator”, you should be able to complete it without a
calculator.
1. Explain how to use a calculator to find lim
x→1
x −1
. Then find the limit.
x2 −1
 1
if x < −1
 x2 ,

if − 1 ≤ x < 1
 2,

if x = 1 . Determine the following limits:
2. Consider the function f ( x ) =  3,
 x + 1,
if 1 < x ≤ 2

 −1 ,
if x > 2
 ( x − 2 )2

b. lim− f ( x )
c. lim f ( x )
d. lim+ f ( x )
e. lim− f ( x )
f. lim f ( x )
g. lim+ f ( x )
h. lim− f ( x )
i. lim f ( x )
lim f ( x )
k. lim f ( x )
l. lim f ( x )
a.
lim f ( x )
x→−1+
x→1
x →2
j.
x→−3
x→−1
x→−1
x→1
x→1
x →2
x →2
x→5
x→1.5
3. Let f be the function whose graph is shown at right.
Calculate each of the following limits or explain why
they do not exist.
a. lim f ( x )
x →0
b. lim f ( x )
x→−1
c. lim− f ( x )
x→1
d. lim f ( x )
x →2
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AP CALCULUS BC – PACKET FOR UNIT 1 LIMITS AND CONTINUITY
4. Find each limit. Simplify your answer.
x 4 − 81
x →2 2 x 2 − 5 x − 3
3− x +5
x →4
x−4
a. lim
b. lim
x2
d. lim
x→−3 3 − x
e. lim
x→−4
x3 − 7 x
x3
f. lim
x−4
x−4
x →0
6
( −4 − x )
c. lim
2
x →4
5. For what values of x is the function continuous?
a.
f ( x) =
x 2 + 3x + 5
x 2 + 3x − 4
 x −1
if x > 1
 x −1 ,

b. f ( x ) =  5 − 3 x, if − 2 ≤ x ≤ 1
 6

,
if x < −2
 x − 4
6. Determine all values of the constant A so that the following function is continuous for all values of x.
 A2 x − A, if x ≥ 3
f ( x) = 
if x < 3
 4,
7. Consider the function f ( x ) =
x2 − 2
. How should f ( x ) be defined at x = 2 to be continuous
x4 − 6x2 + 8
there?
8. Use the Intermediate Value Theorem to determine whether f ( x ) = x5 + 4 x − 1 has a real zero between x = 0
and x = 1 .
9. Use a limit to find the slope of the tangent line to f ( x ) =
x +1
at x = 1 . (Try both methods.)
2− x
10. Use a limit to find the slope of the tangent line to f ( x ) =
1
at the point ( −3, f ( −3) ) . Then write the
−1 − 4 x
equation of this tangent line.
11. A grape is thrown into the air at a velocity of 75 feet per second. Its height (in feet) after t seconds can be
t ) 75t − 16t 2 . Use a limit to find the grape’s instantaneous velocity after 3
described by the function f (=
seconds.
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AP CALCULUS BC – PACKET FOR UNIT 1 LIMITS AND CONTINUITY
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