Calculus Quiz #4 Study Guide (2.1 to 2.3)

Calculus: Quiz #3

(2.1-2.3)

Increasing Function- Continuous rise as x goes from left to right through an interval

-          Decreasing function is the opposite

-          When increasing at a point values are becoming less negative

Extremum (relative extreme point)- relative max or min (described by x); absolute max or min

(described by y)

Changing Slope- change in steepness of slope, increased means higher slope; decreased means

lower slope; when slope is negative, the less negative it gets is increased

Concavity- concave up if slope increases as moved from left to right if there f(x) lies above tan

line

Inflection Point- where function is continuous and graph changes concavity

Describing Graphs

1.      Intervals increase/decrease

2.      Max and min (relative and absolute)

3.      Concave up/down

4.      Inflection points

5.      Undefined points

6.      Asymptotes

First Derivative Rule- If f’(a) >0, then f(x) is increasing at f=a

-          A function is increasing whenever value of its derivative is positive and vice versa

Second Derivative Rule- If f’’(a)>0 then f(x) is concave up at x=a and vice versa

Derivatives	F(x) at x=a
1.      F’(a) is positive 2.      F’’(a) is positive	1.      F(x) is increasing 2.      F(x) is concave up
1.      F’(a) is positive 2.      F’’(a) is negative	1.      F(x) is increasing 2.      F(x) is concave down
1.      F’(a) is negative 2.      F’’(a) is positive	1.      F(x) is decreasing 2.      F(x) is concave up
1.      F’(a) is negative 2.      F’’(a) is negative	1.      F(x) is decreasing 2.      F(x) is concave down

Connection between f(x) and f’(x): y values of f’(x) are slopes of f(x)

            When f(x) is increasing, f’(x) is above x-axis [positive slope]

            When f(x) is decreasing, f’(x) is below x-axis [negative slope]

Four Steps of Curve Sketching

1.      Starting with f(x), compute f’(x) and f’’(x)

2.      Locate relative max and min points and do partial sketch

3.      Find concavity and inflection points

4.      Consider other elements and complete graph

Find relative extremes by setting f’(x) =0

For f(a), a is called the critical number; (a, f(a)) is the critical point

First derivative Test for Local Extreme Points

1.      f’(a)=0

2.      If f’ changes from positive to negative at x=a, f has a local max at a

3.      If f’ changes from negative to positive at x=a, then f has a local min at a

4.      If f’ does not change signs at a, then f has no local extremum at a

Second Derivative Test for Local Extreme Points

1.      If f’(a)=0 and f’’(a)<0, then f has a local max at a

2.      If f’(a)=0 and f’’(a)>0, then f has a local min at a

Solving For Inflection Points

            Set f’’(x)=0 and solve for x

To find if critical point is a relative max or min use first derivative test (number line pos/neg)