Calculus questions

Question

This would be very simple as we are not allowed calculators but i was just wondering how to answer these sorts of questions if asked

 

Answer 1

(1).

The solution as shown below.

Answer 2

(2).

The solution as shown below.

Answer 3

(3).

There are no points at which f " (x) = 0, but at x = ± 2 the function f is not continuous, so test for concavity in the intervals (- ∞, - 2), (- 2, 2) and (2, ∞) as shown in the table.

Test intervals    x - Value       Sign of f " (x)          Conclusion

(- ∞, - 2)                x = - 3             f " (- 3) > 0        Concave upward

(- 2, 2)                   x = 0               f " (0) < 0           Concave downward

(2, ∞)                    x = 3                f " (3) > 0          Concave upward

Answer 4

(7). Power rule of integration :

(8). Integration of trigonometric function (sine) :

(9). Integration of trigonometric function (cosine) :

(10). Integration of natural exponential function (e^x) :

Answer 5

(6).

The solution as shown below.

Answer 6

(4).

Relative (Local) extrema :

• We say that f(x) has a relative (or local) maximum at x = c if f(x) ≤ f(c) for every x in some open interval around x = c.
• We say that f(x) has a relative (or local) minimum at x = c if f(x) ≥ f(c) for every x in some open interval around x = c.

Answer 7

(5).

Absolute (or global) extrema :

• We say that f(x) has an absolute (or global) maximum at x = c if f(x) ≤ f(c) for every x in the domain we are working on.
• We say that f(x) has an absolute (or global) minimum at x = c if f(x) ≥ f(c) for every x in the domain we are working on.