Final Review

Question 1

Derivative of sin(x)

Answer 1

cos(x)

Question 2

Derivative of cos(x)

Answer 2

-sin(x)

Question 3

Derivative of tan(x)

Answer 3

sec^2(x)

Question 4

Derivative of sec(x)

Answer 4

sec(x)tan(x)

Question 5

Derivative of csc(x)

Answer 5

-csc(x)cot(x)

Question 6

Derivative of cot(x)

Answer 6

-csc^2(x)

Question 7

Derivative of arcsin(x)

Answer 7

1/sqrt(1-x^2)

Question 8

Derivative of arccos(x)

Answer 8

-1/sqrt(1-x^2)

Question 9

Derivative of arctan(x)

Answer 9

1/(1+x^2)

Question 10

Derivative of arccot(x)

Answer 10

-1/(1+x^2)

Question 11

Derivative of arccsc(x)

Answer 11

-1/(abs(x)sqrt(x^2-1))

Question 12

Derivative of arcsec(x)

Answer 12

1/(abs(x)sqrt(x^2-1))

Question 13

Derivative of log(x) base a

Answer 13

1/(x*ln(a))

Question 14

Derivative of a^x

Answer 14

a^x * ln(a)

Question 15

Derivative of e^x

Answer 15

e^x

Question 16

Derivative of ln(x)

Answer 16

1/x

Question 17

Extreme value theorem.

Answer 17

If a function f is continuous on a closed interval [a,b], then f has an absolute maximum and absolute minimum on [a,b]

If a function f has a local maximum or minimum at a number c, then either f’(c)=0 or f’(c) does not exist.

Critical numbers are where f’(x)=0 or where f’(x) does not exist.

Question 18

Mean value theorem.

Answer 18

Let f be a function defined on a closed interval [a,b].
If f is continuous on [a,b] and differentiable on (a,b), then there is at least one number c in the open interval (a,b) for which f’(c)=(f(b)-f(a))/(b-a)

Instantaneous rate of change of f at c equals the average rate of change.
The slope of the tangent line equals the slope of the secant line.

Question 19

Intermediate value theorem.

Answer 19

If f is continuous, and f(a)<w<f(b), w=f(c), then there is a value c where f(c)=w.