Local Linear Approximation - crunchy math

Local Linear Approximation

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CD Let fb~ the function given by !(x) = X2 - 2x + 3 The tangent line to the graph off at x = 2 is used to

:~rtoxlmattel.values of~x). ~~h of the following is the greatest value of x for which the error resulting from

angen me approximanon IS less than 0.5 ? .

a. 2.4 b. 2.5

c. 2.6 d. 2.7 e. 2.8

@The function / is twice differentiable with /(2) = I. 1'(2) = 4. and /"(2) = 3. What is the vaJur of the

=\ :approximation of /(1.9) using the line tangent to the graph of / at x 2 ?

I(A) 0.4 (B) 0.6 (C) 0.7 (D) 1.3 (E) 1.4

Students are also expected to know if a tangent line approximation is greater than or less than the
actual function value. This can be determined by the concavity of the original function.

Graph of fis concave down on the interval Graph off is concave up on the interval containing
containing the point of tangency, the tangent line the point of tangency, the tangent line lies below
lies above the curve. the curve.

The tangent line approximation is greater than the The tangent line approximation is less than the

actual value. actual value.

2002AB6 ®

x -1.5 -1.0 -0.5 0 0.5 1.0 1.5

j(x) -1 -4 -6 -7 -6 -4 -1

1'(x) -7 -5 -3 0 3 5 7

Let f be a function that is differentiable for all real numbers. The table above gives the values of f and its

l'derivative for selected points x in the closed interval -1.5:-:; x :-:1; .5. The second derivative of f has

the property that f"(x) > 0 for -1.5 s x s 1.5.

f: 4)S
(a) Evaluate (3j'(x) + dx. Show the work that leads to your answer. .i

(b) Write an equation of the line tangent to the graph of f at the point where x = 1. Use this line to

approximate the value of f(1·2). Is this approximation greater than or less than the actual

value of f(1.2)? Give a reason for your answer.

(c) Find a positive real number r having the property that there must exist a value c with

0< c < 0.5 and f"(e) = r. Give a reason for your answer.

=(d) Let 9 be the function given by g(x) 2X2 - x - 7 for x < 0

{ 2X2 + x - 7 for x 2 O. The graph of 9 passes through

each of the points (x, f(x») given in the table above. Is it possible thatf and 9 are the same

function? Give a reason for your answer.

2008AB 2 Be 2

t (hours) 0 134 7 8 9
L(t) (people) 120 156 176 126 150 80 0

Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours. The number of people
waiting in line to purchase tickets at time t is modeled by a twice-differentiable function L for 0 ~ t ~
, Values of L(t) at various times t are shown in the table above.

(a) Use the data in the table to estimate the rate at which the number of people waiting in line was

=changing at 5:30 p.m. (t 5.5). Show the computations that lead to your answer. Indicate

units of measure.

(b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting
in line during the first 4 hours that tickets were on sale.

(c) For 0 ~ t ~ 9, what is the fewest number of times at which L' (t) must equal O? Give a reason

for your answer.

(d) The rate at which tickets were sold for 0 ~ t ~ 9 is modeled by r'(r) = 550te-t/2 tickets per

=hour. Based on the model, how many tickets were sold by 3 p.m. (t 3), to the nearest whole

number?

Ap® CALCULUS AB 2002 SCORING GUIDELINES

Question 6

x -1.5 -LO -0.5 0 0.5 1.0 1.5

f(x) -1 -4 -6 -7 -6 -4 -1

f'(x) -7 -5 -3 0 3 5 7

Let f be a function that is differentiable for all real numbers. The table above gives the values of f and its
derivative I' for selected points x in the closed interval -1.5 ~ x ~ 1.5. The second derivative of f has the
property that f"(x) > 0 for -1.5 ~ x ~ 1.5.

Jo +(a) Evaluate rl.fi ( 3f' (x) 4) dx. Show the work that leads to your answer.

(b) Write an equation of the line tangent to the graph of f at the point where x = 1. Use this line to
.I
approximate the value of f(1.2). Is this approximation greater than or less than the actual value of f(1.2)?

Give a reason for your answer.

(c) Find a positive real number r having the property that there must exist a value c with 0 < c < 0.5 and

f"(c) = r. Give a reason for your answer.

2X2 - x - 7 for x < 0

Let 9 be the function given by g(x) = +,
!(d)
2X2 X - 7 for x ;:::O.

(x, f(x»)The graph of 9 passes through each of the points given in the table above. Is it possible that f and

9 are the same function? Give a reason for your answer.

ioU(a) 101."( 3f'(x) + 4 )dx = 31015 f'(x) dx + 4 dx antiderivative
2 { 1:
= 3f(x) + 4xll.G = 3(-1- (-7) + 4(1.5) = 24
1: answer
o
tangent line
(b) y=5(x-1)-4
3( ~ : computes y on tangent line at x = 1.2
f(1.2) r::o 5(0.2) - 4 = -3
1: answer with reason
The approximation is less than f(1.2) because the
graph of f is concave up on the interval 2(1 : reference to MVT for f' (or differentiability
1 < x < 1.2. of 1')
1: value of r for interval 0 :S x :S 0.5
(c) By the Mean Value Theorem there is a c with

o < c < 0.5 such that

f"(c) = /'(0.5) - /'(0) = 3 - 0 = 6 = r

0.5 - 0 0.5

(d) lim g'(x) = lim (4x -1) = -1
X""'" 0-
X~O-

lim g'(x) = lim (4x + 1) = +1 2(1 : answers "no" with reference to
x-to+ x_O+ g' or s"
1 : correct reason
Thus g' is not continuous at x = 0, but I' is

=continuous at x 0, so / c;C g.

OR

g"(x) = 4 for all x cc 0, but it was shown in part

=(c) that f"(c) 6 for some c ~ 0, so / c;C g.

Copyright © 2002 by College Entrance Examination Board. All rights reserved.
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

7

Ap® CALCULUS AB
2008 SCORING GUIDELINES

Question 2

t (hours) 0 134789

L(t) (people) 120 156 176 126 150 80 0

Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours. The number of people waiting in

line to purchase tickets at time t is modeled by a twice-differentiable function L for 0 $ t $ 9. Values of L( t) at

various times t are shown in the table above.

(a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at

=5:30 P.M. (t 5.5). Show the computations that lead to your answer. Indicate units of measure .
.I

(b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during
the first 4 hours that tickets were on sale.

(c) For 0 :::;t :::;9, what is the fewest number of times at which L'(t) must equal 0 ? Give a reason for your answer.

°(d) The rate at which tickets were sold for $ t $ 9 is modeled by r( t) = 550te -/ /2 tickets per hour. Based on the
=model, how many tickets were sold by 3 P.M. (t 3), to the nearest whole number?

, L(7)-L(4) = 150-126 = 8 people per hour I: estimate
(a) L (5.5) "" 7 _ 4 3 2: { 1 : units

(b) The average number of people waiting in line during the first 4 hours is I: trapezoidal sum
2 : { 1 : answer
1(approximately
L(O) + L(I) (1 _ 0) + L(I) + L(3) (3 _ 1) + L(3) + L( 4) (4 _ 3)) I: considers change in
.., sign of L'
42 2 2
.J'
= 155.25 people
. { 1: analysis
(c) L is differentiable on [0,9] so the Mean Value Theorem implies 1 : conclusion

°L'(t) > for some t in (1,3) and some t in (4, 7). Similarly,

°L'(t) < for some t in (3,4) and some t in (7,8). Then, since L' is
°continuous on [0,9], the Intermediate Value Theorem implies that

L'(t) = for at least three values of t in [0,9].

OR OR

The continuity of L on [1,4] implies that L attains a maximum value I: considers relative extrema
3 : of L on (0,9)
there. Since L(3) > L(I) and L(3) > L( 4), this maximum occurs on
{ 1 : analysis
= °(1,4). Similarly, L attains a minimum on (3, 7) and a maximum on 1 : conclusion
(4,8). L is differentiable, so L'(t) at each relative extreme point

on (0,9). Therefore L'(t) = 0 for at least three values of tin [0,9].

= °[Note: There is afunction L that satisfies the given conditions with I: integrand
2: { I : limits and answer
L' (t) for exactly three values of t.]

f:(d) r(t) dt = 972.784

There were approximately 973 tickets sold by 3 P.M.

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