Curve Sketching

Question 1

Absolute (global) points

Answer 1

f(x) has a absolute maximum in the domain D at x = c if f(x) </= f(c) for all x in D, and f(x) has an absolute minimum in the domain D at x = c if f(x) >/= f(c) for all x in D.

Question 2

Local (relative) points

Answer 2

f(x) has a local maximum in the domain D at x = c if f(x) </= f(c) for all x in D in some open interval with c and f(x) has a local minimum in the domain D at x = c if f(x) >/= f(c) for all x in D in some open interval with c.

Question 3

EVT

Answer 3

If f is continuous on a closed interval [a,b], then f attains both an absolute maximum and minimum value in [a,b]. If f has a local maximum/minimum at an interior point at a point c in its domain and f’ is defined at c, then f’(c) = 0.

Question 4

Interior points

Answer 4

Every point in a closed interval that does not include the end/boundary points.

Question 5

Critical points

Answer 5

An interior point in the domain of a function f where f’ is zero or undefined. Can be discontinuous points, continuous points that are not differentiable and differentiable points whose derivative = 0.

Question 6

Finding absolute extrema of a continuous function on a closed interval

Answer 6

1. Find all critical points of f on the interval. 2. Evaluate f at all critical and end points. 3. Take the largest and smallest values for the absolute maximum/minimum.

Question 7

First derivative test

Answer 7

Use if x = c is a continuous critical point. 1. if f’ changes from positive to negative at x = c, then x = c is a local minimum. 2. if f’ changes from negative to positive at x = c, then x = c is a local maximum. 3. If the sign does not change, there is no local extreme.

Question 8

Inflection points

Answer 8

A point (c, f(c)) where the graph of a function has a tangent line and where concavity changes. 1. The tangent line must exist. 2. The concavity much change at x = c. 3. x = c is an inflection point if x = c is continuous and f’(c) is a finite number or +/- INF. 4. f’‘(c) changes sign at x = c.

Question 9

Second derivative test

Answer 9

Use if f’’ is continuous on an open interval with x = c. 1. If f’(c) = 0 and f’‘(c) < 0 (concave down, negative), then f has a local maximum at x = c. 2. If f’(c) = 0 and f’‘(c) > 0 (concave up, positive), then f has a local minimum at x = c. 3. If f’(c) = 0 and f’‘(c) = 0, then then test fails.

Question 10

Curve Sketching

Answer 10

Find critical points, local and absolute extrema, inflection points, and then sketch. Note asymptotes and x/y intercepts.