Page 82 - Calculus Demystified

CHAPTER 2
Foundations of Calculus
We conclude that the instantaneous velocity of the moving body at time t = 5 69
is g (5) = 115 ft/sec.

Math Note: Since position (or distance) is measured in feet, and time in seconds,
then we measure velocity in feet per second.

EXAMPLE 2.12
3
Calculate the slope of the tangent line to the graph of y = f(x) = x − 3x
at x =−2. Write the equation of the tangent line. Draw a figure illustrating
these ideas.

SOLUTION

We know that the desired slope is equal to f (−2). We calculate
f(−2 + h) − f(−2)

f (−2) = lim
h→0 h
3 3
[(−2 + h) − 3(−2 + h)]−[(−2) − 3(−2)]
= lim
h→0 h
2 3
[(−8 + 12h − 6h + h ) + (6 − 3h)]+[2]
= lim
h→0 h
2
3
h − 6h + 9h
= lim
h→0 h
2
= lim h − 6h + 9
h→0
= 9.
3
We conclude that the slope of the tangent line to the graph of y = x − 3x at
x =−2 is 9. The tangent line passes through (−2,f(−2)) = (−2, −2) and
has slope 9. Thus it has equation

y − (−2) = 9(x − (−2)).

The graph of the function and the tangent line are exhibited in Fig. 2.9.

2
You Try It: Calculate the tangent line to the graph of f(x) = 4x − 5x + 2atthe
point where x = 2.

EXAMPLE 2.13
A rubber balloon is losing air steadily. At time t minutesthe balloon contains
2
75− 10t + t cubic inchesof air.What isthe rate of lossof air in the balloon
at time t = 1?