Test 3

Question 1

Concavity:

Answer 1

Refers to the way the graph bends.

Question 2

Concave Up:

Answer 2

F’ is increasing, F’’ is positive.

Question 3

Concave Down:

Answer 3

F’ is decreasing, F’’ is negative.

Question 4

Points of Inflection:

Answer 4

The concavity changes from up to down or vice versa.
F’’ is 0 or undefined.

Question 5

If F’’ goes from positive to negative:

Answer 5

F’ has a local max.

Question 6

If F’’ goes from negative to positive:

Answer 6

F’ has a local min.

Question 7

F’’ > 0 then,

Answer 7

F is a local min.

Question 8

F’’ < 0 then,

Answer 8

F is a local max.

Question 9

L’Hopital’s Rule:

Answer 9

If f(x)/g(x) has an indeterminate form, we can replace it with f’(x)/g’(x).

Question 10

Growth of e^x:

Answer 10

as lim x->infinity:
e^x/x^n = infinity.

Question 10

Delta X:

Answer 10

(b-a)/N
b - right endpoint of interval
a - left endpoint of interval
N - number of intervals.

Question 11

Right Point Approximation:

Answer 11

Delta X [j=1; N, f(xj)]

Question 12

Left Point Approximation:

Answer 12

Delta X [j=0; N-1, f(xj)]

Question 13

Mid Point Approximation:

Answer 13

Delta X [j=0; N-1, {(xj+xj+1)/2}]

Question 14

Riemann Sums:

Answer 14

The rectangles do not need to have equal width, f(x) can have any real value, and the height can be any value of f(x) within the subinterval.

Question 15

Riemann Sums - P, N, and C

Answer 15

P - partition
N - choice of points that divides it into N subintervals.
C - sample points.

Question 16

||P|| (Riemann Sums)

Answer 16

The maximum of the lengths of delta X.

Question 17

The Definite Integral and Signed Area:

Answer 17

Allowing the possibility that f(x) takes on both positive and negative values, we define the notion of the area below the x-axis as providing a negative contribution.

Question 18

Reversing the Limits of Integration:

Answer 18

If the limits are opposite, you can reverse them and make everything negative.

Question 19

Power Rule for Integrals:

Answer 19

[x^(n+1)] / (x+1)

Question 20

Antiderivative of (dx / x):

Answer 20

ln |x| + C

Question 21

Sum and Constant Rules of Antiderivatives:

Answer 21

Apply the same as with derivatives.

Question 22

Antiderivative of e^x:

Answer 22

e^x + C

Question 23

Antiderivative of e^kx:

Answer 23

(1 / k) * e^(kx) + C

Question 24

Fundamental Theorem of Calculus Part 1:

Answer 24

For the antiderivative of f(x) from a to b, the answer to the problem is:

F(b) - F(a)

Question 25

The Fundamental Theorem of Calculus Part 2:

Answer 25

Demonstrates how definite integration and differentiation are inverse processes.

Question 26

Net Change as the Integral of a Rate of Change:

Answer 26

The integral of velocity is equal to the net change in position.
The distance traveled would be the absolute value.