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Its All about Pj Problem Strings - 7 Spaces Of Interest and their associated Basic Sequences; 7 Pj Problems of Interest (PPI) and their Alleles (A)

Single Variable Functions - Limits, Differentiability, Continuity

Calculus plays a major role in the search for solutions for myriad existential problems. Limits, differentiability and continuity are some of the central concepts in calculus.
Limits Theorems

1. Consider the general polynomial p(x) = a0 + a1x + a2x2 + ... + anxn.
Show that limitx→a p(x) = p(a). Where a is a real number.
Ans:1. limitx→a p(x) = limitx→a (a0 + a1x + a2x2 + ... + anxn) = p(a) (after applying limits theorems)
In general for any polynomial p, limitx→a p(x) = p(a)
Also, if a rational function r(x) = p(x)/q(x); where p and q are polynomials and q(x) ≠ 0;
Then, limitx→a r(x) = r(a)

2. Determine the following limits:
(a) limx→4 (5x2 - 2x + 3)
(b) limt→-2 (t + 1)9(t2 - 1)
(c) limt→4- √(16 - x2)
(d) limt→-3 [(x2) - x + 12)]/(x + 3)
Ans:2(a) f(4) = 75
(b) f(-2) = -3
(c) f(4) = 0
(d) Limit does not exist because as x tends to -3, x + 3 tends to 0 eventhough (x2 - x + 12) tends to 24. There is no way to simplify this ratio so as to remove the denominator. Sometimes such removal is possible. For example, if the numerator had -12 instead of +12, this would have been possible through factorization.

3(a) Write the difference quotient for f(x).
(b) Write the difference quotient for f(x) at x = a.
(c) What is another name for the limit of the difference quotient for f(x) at x = a as the denominator tends to 0?
(d) What is the geometric interpretation of the answer to 3(c)?
(e) What is the interpretation of the answer to 3(c) with respect to rate of change?
Ans:3(a) [f(x + h) - f(x)]/h. Where h ≠ 0.
(b) [f(a + h) - f(a)]/h.
(c) limh→0 [f(a + h) - f(a)]/h is also called the derivative of f at a and is denoted as f'(a). f is differentiable at a if the derivative f'(a) exists.
(d) The derivative f'(a) is gometrically interpreted as the slope of the tangent to f(x) at a.
(e) The derivative f'(a) is also interpreted as the instantaneous rate of change of f with respect to x at a.

4. Use the difference quotients of the following functions to determine their derivatives:
(a) f(x) = x2.
(b) f(x) = 1/x.
Ans:4(a) Simplification of the difference quotient results in
2x + h. So, limh→0 2x + h = 2x.
Therefore derivative = f'(x) = 2x
(b) Simplification of the difference quotient results in
-1/(x(x + h)). So, limh→0 -1/(x(x + h))
Therefore derivative = f'(x) = -1/x2.

5(a) Relate differentiability with continuity
(b) Relate Limits with continuity.
Ans:5(a). If f(x) is differentiable at x = a, then f(x) is continuous at x = a. Although differentiability implies continuity, continuity does not imply differentiability. In other words, the continuity of f(x) at x = a, does not imply that f(x) is differentiable at x = a.
(b) If limx→a f(x) = f(a), then f(x) is continuous at x = a.

6. f(x) = (x2 -x - 6)/(x - 3)
(a) Is f(x) continuous at x = 3?
(b) Is f(x) differentiable at x = 3?
Ans:6(a) limh→0 f(x) = 5. f(3) = 4. So f(x) is not continuous at x = 3.
(b) If f(x) is not continuous at x = 3, it is not differentiable at x = 3.

7. If f(x) = x2 - x is continuous on ℜ What is the limit of g(x) = ef(x) as x tends to 1?
Ans: If f(x) is continuous on ℜ, then g(x) is continuous on ℜ. So limx→1 g(x) = g(1) = 1

8. Determine if f(x) is increasing or decreasing at x = a if:
(a) f'(a) > 0
(b) f'(a) < 0
(c) f'(a) = 0.
Ans: 8(a) if f'(a) > 0, then f(x) is increasing at x = a (first derivative rule)
(b) If f'(a) < 0, then f(x) is decreasing at x = a (first derivative rule)
(c) Not clear whether f(x) is increasing or decreasing, more information is needed.

9. Determine if f(x) is concae up or concave down at x = a if:
(a) f''(a) > 0
(b) f''(a) < 0
(c) f'(a) = 0.
Ans: 9(a) if f''(a) > 0, then f(x) is concave up at x = a (second derivative rule)
(b) If f''(a) < 0, then f(x) is concave down at x = a (second derivative rule)
(c) Not clear whether f(x) is convave up or concave down, more information is needed.

10. Evaluate limx→∞ sin(1/x).
Ans: 0.

Single Variable Functions: Domains
Single Variable Functions: Limits. Differentiability And Continuity
Single Variable Functions The Derivative As Rate Of Change
Conics
Vector Spaces
Real Numbers