# Calculus I exam 1

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1. Deffiniation of a Limit

-Let f be a function defined on an open interval containing c (exept possibly at c) and let L be a real number
$\lim_{x \to c}f(x) = L$
means that f(x) can be made to be as close to L as desired by making x sufficiently close to c. In that case, it can be stated that "the limit of f of x, as x approaches c, is L". Note that this statement can be true even if f(c) ≠ L. Indeed, the function f(x) need not even be defined at c.
2. commontypes of behaviors associated with the non-exitence of a limit
• 1. f(x) approaches a different number from the right side of c than it approaches from the left side.
• 2. f(x) increases or decreases without bound as x approaches c
• 3. f(x) oscillates between two fixed values as x approaches c
3. What are three basic limits
$\lim_{x \to c} a = a$
$\lim_{x \to c} x = c$
$\lim_{x \to c} x^r = c^r \qquad \mbox{ if } r \mbox{ is a positive integer}$
4. Properties of limits: Sum or difference
$\text{If }\lim_{x \to c} f(x) = L_1 \text{ and }\lim_{x \to c} g(x) = L_2 \text{ then:}$
$\lim_{x \to c} \, [f(x) \pm g(x)] = L_1 \pm L_2$
5. Properties of limits: Product
$\text{If }\lim_{x \to c} f(x) = L_1 \text{ and }\lim_{x \to c} g(x) = L_2 \text{ then:}$
$\lim_{x \to c} \, [f(x)g(x)] = L_1 \times L_2$
6. Properties of limits: Quotient
$\lim_{x\to c} f(x) =L$ and $\lim_{x\to c} g(x) =M$
• $\lim_{x\to c} \frac{f(x)}{g(x)} = \frac{\lim_{x\to c} f(x)}{\lim_{x\to c} g(x)} = \frac{L}{M} \,\,\, \mbox{ provided } M\neq 0$

7. Properties of limits: Power
$\lim_{x\to c} f(x) =L$ and $\lim_{x\to c} g(x) =M$
• $\lim_{x\to c} [f(x) g(x)] = \lim_{x\to c} f(x) \lim_{x\to c} g(x) = L M$

8. Limit of a composite Function

If f is continuous at b and lim x→a g(x) = b
then lim x→a f(g(x)) = f (lim x→a g(x)) = f (b).
9. Limit of Trigonometric Functions:
find limx→c for sin(x) ; cos (x) ; sin(x)/x ; (1-cos(x))/(x)
10. Example : Evaluate
• Substituting 0 for x yields 5/0, which is meaningless; hence, = DNE. (Remember, infinity is not a real number.)
• THIS IS AN ASYMPTOTE, NON-REMOVABLE DISCONTINUITY (numerator ≠ 0 , denominator = 0)
11. A stradegy For finding limits
• 1. Learn to recognize which limits can be evaluated by direct substituation
• 2. If the limit of f(x) as x approaches c cannot be evaluated by direct substituaiton, try to find a afunction g that agrees with f for all of x other than x=c
• 3. apply the theorem for removable discontinuities such that; lim x→c f(x) = lim x→c g(x) = g(c)
• 4. use a graph or table to reinforce conclusion
12. Functions that agree at all but one point

let c be real and let f(x) = g(x) for all x ≠ c in an open interval containing c
If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists

lim x→c f(x) = lim x→c g(x)
13. Definition of Continuity
• 1. f(c) is defined
• 2. lim x→c f(x) exists
• 3. $\lim_{x \to c}{f(x)} = f(c).$

• continuity on an open interval: a function is continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line ( $-\infty$ , $+\infty$ ) is everywhere continuous.
14. The existence of a limit

let f be a function and let c and L be real numbers
$\lim_{x \to c}f(x) = L$ if and only if $\lim_{x \to p^+}f(x) = L$ and $\lim_{x \to p^-}f(x) = L$
15. Definition of continuity on a closed interval
A function is said to be continuous on [a,b] if and only if

• 1. it is continuous on (a,b),
• 2. it is continuous from the right at a and
• 3. it is continuous from the left at b.
16. properties of continuity

1. scalar multiple:
2. sum or difference:
3. Product:
4. quotient:
• 1. scalar multiple: bf
• 2. sum or difference: f (+ - ) g
• 3. Product: fg
• 4. quotient: f/g, if g(c) ≠ 0
17. Continuity of a compsoite function
If the function g is continuous at a number a, and f is continuous at g(a), the composite function is also continuous at a.
18. intermediate value theorem
If a function f is continuous on a closed interval [a,b], then for every value k between f(a) and f(b) there is a value c between a and b such that f(c) = k.
19. definition of infinite limits
We say if we can make f(x) arbitrarily large for all x sufficiently close to x=a, from both sides, without actually letting .

We say if we can make f(x) arbitrarily large and negative for all x sufficiently close to x=a, from both sides, without actually letting .
20. Definition of a vertical asymptote
if f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x=c is a vertical asymptote of the graph of f
21. finding vertical asymptote
• let h and g be continuous on an open interval containing c. If g(c) ≠ 0, h(c) = 0, and there exists an open interval containing c such that g(x) ≠ 0 for all x ≠ c in the interval, then the graph of the function given by:
• has a vertical asymptote at x = c
22. properties of infinite limits: sum or difference
Given the functions and suppose we have,

for some real numbers c and L.
23. Properties of Infinite Limits: Product
Given the functions and suppose we have,

for some real numbers c and L. Then,
then

then
24. Properties of infinite limits: Quotient

Given the functions and suppose we have,

for some real numbers c and L. Then,
25. Definition of a tangent line with slope m

also known as the slope of the graph of f at x = a
if f is defined on an open interval containing a, and if the limit: $\frac{f(a+h)-f(a)}{h}.$ exists, then the lines passing through (a, f(a)) with slpoe m is the tangent line to the graph of f at the point (a, f(a))
26. Definition of the derivative of a function
the derivative of f at x
$m = \frac{\Delta f(x)}{\Delta x} = \frac{f(x+h)-f(x)}{h}.$
27. differentialbility implies continuity
if f is differentiable at x = c, then f is continuous at f = c
28. The constant rule of the derivative
the derivative of a constant function is 0. That is, if c is a real number, then
29. the power rule of the derivative
if n is a rational number, then the function f(x) = xn is differentialble. For f to be differentiable at x=0 then n must be a number such that xn-1 is defined on the interval containing 0

30. The sum and differnece rules of the derivative
• The derivative of the sum of two functions is the sum of the derivatives of the two functions:
• .
• Likewise, the derivative of the difference of two functions is the difference of the derivatives of the two functions.
31. the derivative of the sin and cos function
32. Alternate limit form of the derivative
33. How to find the equation of the tangent line through a point (x1,y1)
find the derivative f'(c) = m = slope

plug m into the point slope formula y-y1=m(x-x1)
 Author: ithuestad ID: 106908 Card Set: Calculus I exam 1 Updated: 2011-10-07 15:44:29 Tags: Calculus limits tangent secant Folders: Description: Chapter 1 and sections 2.1 and 2.2 Show Answers: