Answer to Question #155472 in Calculus for Tom Garland

Question

Let f be a function such that each point (x,y) on the graph of f, the slope given by dy/dx = y^2-x. The graph of f passes through the point (1,2) and is concave down on the interval 1<x<1.5. Let k be the approximation of f (1.2) found by using the locally linear approximation of f at x=1. Which of the following statements about k is true?

a) k=5.6 and is an overestimate of f(1.2)

b) k=5.6 and is an underestimate of f(1.2)

c) k-2.6 and is an overestimate of f(1.2)

d) k=2.6 and is an underestimate of f(1.2)

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 (c) is true.