*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)*

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.

1. Consider the general polynomial ** p(x) = a _{0} + a_{1}x + a_{2}x^{2} + ... + a_{n}x^{n}**.

Show that

In general for any polynomial p,

Also, if a rational function r(x) = p(x)/q(x); where p and q are polynomials and q(x) ≠ 0;

Then,

2. Determine the following limits:

(a) **lim _{x→4} (5x^{2} - 2x + 3)**

(b)

(c)

(d)

(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 (x

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) **lim _{h→0} [f(a + h) - f(a)]/h** is also called the

(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) = x^{2}.

(b) f(x) = 1/x.
**Ans:**4(a) Simplification of the difference quotient results in

2x + h. So, **lim _{h→0} 2x + h = 2x**.

Therefore derivative = f'(x) = 2x

(b) Simplification of the difference quotient results in

-1/(x(x + h)). So,

Therefore derivative = f'(x) = -1/x

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 **lim _{x→a} f(x) = f(a)**, then f(x) is continuous at x = a.

6. f(x) = (x^{2} -x - 6)/(x - 3)

(a) Is f(x) continuous at x = 3?

(b) Is f(x) differentiable at x = 3?
**Ans:**6(a) **lim _{h→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) = x^{2} - x is continuous on ℜ What is the limit of g(x) = e^{f(x)} as x tends to 1?
**Ans:** If f(x) is continuous on ℜ, then g(x) is continuous on ℜ. So **lim _{x→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 **lim _{x→∞} sin(1/x).
Ans: 0.
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