Page 92 - Calculus Demystified

Foundations of Calculus
CHAPTER 2
2. Determine whether the given function f is continuous at the given point c. 79
Give careful justifications for your answers.
x − 1
(a) f(x) = c =−1
x + 1
x − 1
(b) f(x) = c = 3
x + 1

(c) f(x) = x · sin(1/x) c = 0
(d) f(x) = x · ln x c = 0

x 2 if x ≤ 1
(e) f(x) = c = 1
x if 1 <x

x 2 if x ≤ 1
(f) f(x) = c = 1
2x if 1 <x

sin x if x ≤ π
(g) f(x) = c = π
(x − π) if π< x
(h) f(x) = e ln x+x c = 2
3. Use the definition of derivative to calculate each of these derivatives.
2

(a) f (2) when f(x) = x + 4x

(b) f (1) when f(x) =−1/x 2
4. Calculate each of these derivatives. Justify each step of your calculation.

x
(a)
2
x + 1
d
2
(b) sin(x )
dx
d 3 2
(c) t · tan(t − t )
dt
2
d x − 1
(d)
2
dx x + 1
(e) [x · ln(sin x)]
d s(s+2)
(f) e
ds