Columbia University Mathematics Worksheet


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10. f(x) = 2×2 – %, a = 5, Ax= -0.4
11. f(x) =
a = 3, Ar=0.5
1 + x2
12. f(x) = ln(x2 + 1), a = 1, Ax = 0.1
In Exercises 13–20, using Linear Approximation, estimate Af for a
change in x from x = a to x = b. Use the estimate to approximate f(b),
and find the error using a calculator.
13. f(x)= Vx, a= 25, b = 26 14. f(x)=x1/4, a=16, b=16.5
15. f(x)= twa=100, b = 101 16. f(x)= ta, a = 100, b=98
17. f(x)=x1/3, a=8, b=9 18. f(x)= tan-1x, a=1, b=1.05
19. f(x)=ex, a=0,b=-0.1 20. f(x)=lnx, a=1, b=0.97
In Exercises 21-28, find the linearization at x = a and then use it to
approximate f(b).
21. f(x) = x4, a=1, b=0.96
22. f(x) =
) = a = 2, b = 2.02
23. f(x) = sin’x, a = 1, b = 147
24. f(x)=
a= 4, b = 4.1
25. f(x) = (1+x)-1/2, a = 0, b= 0.08
26. f(x) = (1 + x)-1/2, a =3, b = 2.88
27. f(x) = evi, a=1, b=0.85
28. f(x) = et Inx, a=1, b = 1.02
In Exercises 29–32, estimate Ay using differentials (Eq. (3)).
29. y = cos X,
a=t, dx = 0.014
34. At a certain moment, an object in linear motion has velocity
100 m/s. Estimate the distance traveled over the next quarter-second, and
explain how this is an application of the Linear Approximation.
35. Which is larger: V2.1 – V2 or V9.1 – V9? Explain using the Linear
36. Estimate sin 61° – sin 60° using the Linear Approximation. Hint:
Express AO in radians.
37. Box office revenue at a cinema in Paris is R(p) = 3600p – 10p3 eu-
ros per showing when the ticket price is p euros. Calculate R(p) for p = 9
and use the Linear Approximation to estimate AR if p is raised or lowered
by 0.5 euro.
38. The stopping distance for an automobile is F(s) = 1.1s +0.054s2 ft,
where s is the speed in mph. Use the Linear Approximation to estimate
the change in stopping distance per additional mph when s = 35 and when
s = 55.
39. A thin silver wire has length L = 18 cm when the temperature is
T = 30°C. Estimate AL when T decreases to 25°C if the coefficient of
thermal expansion is k = 1.9 x 10-50c-1 (see Example 6).
40. At a certain moment, the temperature in a snake cage satisfies
dT/dt = 0.008°C/s. Estimate the rise in temperature over the next 10 s.
41. The atmospheric pressure at altitude h (kilometers) for 11

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