Question

<e> Let f

f be a function such that at each point (x,y)

(x,y) on the graph of f

f, the slope is given by dy

dx

=1

2

x−1

4

y

2

dydx=12x−14y2. The graph of f

f passes through the point (1,−2)

(1,−2) and is concave up on the interval 1<x<1.5

1<x<1.5. Let k

k be the approximation for f(1.3)

f(1.3) found by using the locally linear approximation of f

f at x=1

x=1. Which of the following statements about k

k is true?

Answer 1

Since the function is concave down, it is overestimated and $f''(x)<0$

$y''=2yy'-1$

Then:

for $k=f(1.2)=5.6$ :

$2yy'-1=2\cdot5.6\cdot(5.6^2-1.2)-1>0$

for $k=f(1.2)=-2.6$ :

$2yy'-1=-2\cdot2.6\cdot(2.6^2-1.2)-1<0$

Answer: statement (k-2.6 and is an overestimate of f(1.2)) is true.