SOLUTION: Use the line tangent to at x = 0; f (0) = 3.
, so f ′(0) = 6 ; hence, the line is y = 6x + 3.Our tangent-line approximation, then, is .
At x = 0.05, we have f (0.05) ≈ 6(0.05) + 3 = 3.3.
To determine if this is an overestimate or an underestimate, we examine the
second derivative. for x ≠ 1; therefore, f (x) is always concave
up.
Therefore the tangent line at x = 0 lies below the graph of f (x) on the interval [0,
0.05]. The estimate is on the tangent line, so it must be an underestimate. We can
verify this by evaluating f (0.05) or by graphing f (x) and its tangent line at x = 0
on a calculator.
M. RELATED RATES
If several variables that are functions of time t are related by an equation, we can
obtain a relation involving their (time) rates of change by differentiating with
respect to t.
Example 34 __
If one leg AB of a right triangle increases at the rate of 2 inches per second,
while the other leg AC decreases at 3 inches per second, find how fast the
hypotenuse is changing when AB = 6 feet and AC = 8 feet.
Figure N4–21