Exam 2 Formula Sheet
Content
EXAM 2 - FORMULA SHEET
1. Linearization Any differentiable function f can be approximated by its tangent line at the point a: L(x) = f (a) + f (a)(x − a) 2. Differentials If y = f (x) then the differentials are defined through dy = f (x)dx. 3. Differences The amount of change or increment ∆y of a function y = f (x) for a (small) increment ∆x is given by ∆y = f (x + ∆x) − f (x) 4. Increasing/Decreasing Tests f (x) > 0 on an interval ⇒ f is increasing on that interval. f (x) < 0 on an interval ⇒ f is decreasing on that interval. 5. Concavity Test f ”(x) > 0 on an interval ⇒ f is concave upward on that interval; while f ”(x) < 0 on an interval ⇒ f is concave downward on that interval. 6. A point (c, f (c)) where a curve changes concavity is called an inflection point. If f ” is continuous near c, then c is an inflection point ⇒ f ”(c) = 0. f ”(c) = 0 DOES NOT IMPLY THAT c is an inflection point. Example: The graph of y = x4 has no inflection point at 0 even though f ”(0) = 0. This is because it is concave upwards (f ” > 0) on both sides of 0. 7. Extreme Value Theorem If f is continuous on [a, b], then f attains an absolute maximum value f (c) and an absolute minimum value f (d) at some numbers c and d in [a, b]. 8. A critical number of a function f is a number c in the domain of f such that either f (c) = 0 or f (c) does not exist. 9. Fermat’s Theorem f has a local extremum at c ⇒ c is a critical number of f . 10. Mean Value Theorem If f is a differentiable function on the interval [a, b] then there exists a number c between a and b such that f (c) = f (b) − f (a) b−a
11. The Closed Interval Method Steps to find the absolute max/min of f in a closed interval. • Find the values f (c) at the critical numbers c in [a, b]. • Find f (a) and f (b). • The largest of all these values is the absolute maximum of f and the smallest is the absolute minimum in that interval.
12. The First Derivative Test Suppose that c is a critical number of a continuous function f . (a) f changes from positive to negative at c ⇒ f has a local maximum at c. (b) f changes from negative to positive at c ⇒ f has a local minimum at c. (c) f does not change sign at c ⇒ f does not have a local max/min at c. 13. The Second Derivative Test Suppose f ” is continuous near c. (a) f’(c)=0 and f ”(c) > 0 ⇒ f has a local minimum at c. (b) f’(c)=0 and f ”(c) < 0 ⇒ f has a local maximum at c. 14. The line x = a is called a vertical asymptote of the curve y = f (x) if at least one of the following statements is true: lim f (x) = ∞ lim f (x) = ∞
x→a−
x→a
x→a+
lim f (x) = ∞ lim f (x) = −∞
x→a
lim f (x) = −∞
lim f (x) = −∞
x→a−
x→a+
15. The line y = L is called a horizontal asymptote of the curve y = f (x) if either limx→∞ = L or limx→−∞ = L 16. For any positive integer n,
x→−∞
lim x−n = 0
x→∞
lim x−n = 0
17. The following limit is important:
x→−∞
lim ex = 0
18. Suppose that the values of f (x) become larger and larger as x becomes larger and larger. We write lim f (x) = ∞
x→∞
19. Curve sketching • (a) Find all of the vertical and horizontal asymptotes. • (b) Find the intervals of increase or decrease. • (c) Find the local maximum and minimum values. • (d) Find the intervals of concavity and the inflection points. • (e) Check for any symmetries and whether the function is periodic. • (f) Draw the graph!
1. Linearization Any differentiable function f can be approximated by its tangent line at the point a: L(x) = f (a) + f (a)(x − a) 2. Differentials If y = f (x) then the differentials are defined through dy = f (x)dx. 3. Differences The amount of change or increment ∆y of a function y = f (x) for a (small) increment ∆x is given by ∆y = f (x + ∆x) − f (x) 4. Increasing/Decreasing Tests f (x) > 0 on an interval ⇒ f is increasing on that interval. f (x) < 0 on an interval ⇒ f is decreasing on that interval. 5. Concavity Test f ”(x) > 0 on an interval ⇒ f is concave upward on that interval; while f ”(x) < 0 on an interval ⇒ f is concave downward on that interval. 6. A point (c, f (c)) where a curve changes concavity is called an inflection point. If f ” is continuous near c, then c is an inflection point ⇒ f ”(c) = 0. f ”(c) = 0 DOES NOT IMPLY THAT c is an inflection point. Example: The graph of y = x4 has no inflection point at 0 even though f ”(0) = 0. This is because it is concave upwards (f ” > 0) on both sides of 0. 7. Extreme Value Theorem If f is continuous on [a, b], then f attains an absolute maximum value f (c) and an absolute minimum value f (d) at some numbers c and d in [a, b]. 8. A critical number of a function f is a number c in the domain of f such that either f (c) = 0 or f (c) does not exist. 9. Fermat’s Theorem f has a local extremum at c ⇒ c is a critical number of f . 10. Mean Value Theorem If f is a differentiable function on the interval [a, b] then there exists a number c between a and b such that f (c) = f (b) − f (a) b−a
11. The Closed Interval Method Steps to find the absolute max/min of f in a closed interval. • Find the values f (c) at the critical numbers c in [a, b]. • Find f (a) and f (b). • The largest of all these values is the absolute maximum of f and the smallest is the absolute minimum in that interval.
12. The First Derivative Test Suppose that c is a critical number of a continuous function f . (a) f changes from positive to negative at c ⇒ f has a local maximum at c. (b) f changes from negative to positive at c ⇒ f has a local minimum at c. (c) f does not change sign at c ⇒ f does not have a local max/min at c. 13. The Second Derivative Test Suppose f ” is continuous near c. (a) f’(c)=0 and f ”(c) > 0 ⇒ f has a local minimum at c. (b) f’(c)=0 and f ”(c) < 0 ⇒ f has a local maximum at c. 14. The line x = a is called a vertical asymptote of the curve y = f (x) if at least one of the following statements is true: lim f (x) = ∞ lim f (x) = ∞
x→a−
x→a
x→a+
lim f (x) = ∞ lim f (x) = −∞
x→a
lim f (x) = −∞
lim f (x) = −∞
x→a−
x→a+
15. The line y = L is called a horizontal asymptote of the curve y = f (x) if either limx→∞ = L or limx→−∞ = L 16. For any positive integer n,
x→−∞
lim x−n = 0
x→∞
lim x−n = 0
17. The following limit is important:
x→−∞
lim ex = 0
18. Suppose that the values of f (x) become larger and larger as x becomes larger and larger. We write lim f (x) = ∞
x→∞
19. Curve sketching • (a) Find all of the vertical and horizontal asymptotes. • (b) Find the intervals of increase or decrease. • (c) Find the local maximum and minimum values. • (d) Find the intervals of concavity and the inflection points. • (e) Check for any symmetries and whether the function is periodic. • (f) Draw the graph!
