AP Calc AB (Serrano) 2.1-2.5 Notes
Solve the following: lim(x→∞) AND lim(x→-∞) √(x² + 2)/(3x - 6)
(x→∞): (1/√x²)/(1/x) BECAUSE 1/√x² = 1/x under √ = √(1 + 2/x²)/(3 - 6/x) = √(1 + 0)/(3 - 0) = √1/3 = 1/3 (x→-∞): make the x's negative (1/-x), though 1/√x² is the same
Find lim(t→0) [√(t² + 9) - 3]/t²
**can't apply quotient law immediately, since limit of denominator is 0 ***rationalize the NUMERATOR with conjugate lim(t→0) [√(t² + 9) - 3]/t² * [√(t² + 9) + 3]/[√(t² + 9) + 3] lim(t→0) (t² + 9 - 9)/t² * 1/[√(t² + 9) + 3] 9s cancel out then t² cancels out so = lim(t→0) 1/[√(t² + 9) + 3] substitute in 0; lim(t→0) 1/√9 + 3 = lim(t→0) = 1/(3 + 3) = 1/6
Evaluate lim(x→0⁻) e^(1/x)
**composite function f = 1/x lim(x→0⁻) 1/x = -∞ lim(f→-∞) e^f = 0 as e approaches -∞, it approaches 0
Find the limit of the following: lim(x→∞) sin²x/x²
**squeeze theorem 1/x² has hole at 0 so [0,1] instead of [-1,1] 0 ≤ sin² x ≤ 1 divide each side by x² 0 ≤ sin² x ≤ 1/x²
Describe whether the following indicates approaching a function from the left, right, or both directions: . x→a⁻ . x→a⁺ . x→a
. left . right . both directions
Use a your calculator to find the value of lim (x-1)/(x²-1) x→1___________
0.5 = .66667 0.9 = .52637 0.99 = .50251 0.999 = .50025 1 = ERROR (hole bc 0 in denominator) 1.001 = .49975 1.01 = .49751 1.1 = .47619 the limit = 0.5 bc the graph is approaching it
Given the below piecewise function, is the function continuous at x = 3? f(x) = { x + 7 when x > 3, { 10-x² when x ≤ 3) 1) Is f(3) defined? 2) Does the limit exist? 3) Do they equal each other?
1) 10-(3)² = 1, yes 2) lim(x→3⁻) 10-3² = 1; lim(x→3⁺) 3 + 7 = 10 no because 1 ≠ 10 3) no; 3 does not exist if 1 or 2 do not (remember on tests). f(x) is discontinuous at x = 3 because lim(x→3) f(x) = DNE
Solve lim(x→-2) (x³ + 2x² - 1)/(5-3x)
1) find the domain 5 - 3x = 0; (-∞, 5/3)u(5/3, ∞) -2 ≠ 5/3 so solve normally; direct substitution (-2)³ + 2(-3)² - 1/5 - 3(-2) = -1/11
Evaluate lim(x→∞) √(x² + 1) - x (radical function)
1) rewrite as a rational function √(x² + 1) - x/1 * (√(x² + 1) + x)/(√(x² + 1) + x) 2) divide by highest number in the denominator x in denom so 1/x; change to 1/x² for denominator with radical ( * (1/x)/(1/x²)) = (1/x)/(√((x²/x²) + (1/√x²) + x/x²) = (1/x)/(√(1 + 1/x) + 1) = 0/1 = 0
Find lim(x→0) ln(tan² x)
1) t = tan²x; because t² always positive and ∞; t ≥ 0 lim(x→0) tan²x = 0 (direct substitution) lim(t → 0⁺) ln (t) = -∞; not mathematically possible to take the log of 0 If you visualize y = ln x approaches 0 0⁺ because not defined from the left
Where is f(x) = (ln x) + (tan⁻¹ x)/(x² - 1) continuous?
4 functions; find the domain of each and combine where they overlap y = ln x is continuous at (0,∞) y = tan⁻¹ x is continuous at (-∞,∞) y = x² - 1 is continuous at (-∞,∞) EXCEPT it's in the domain so (-∞,∞) ∋ 1,-1 draw a number line and see where they overlap; domain = (0,1)u(1,∞ so continuous there
If a line has a colored in dot and you are asked to approach it from the left (but it's an endpoint), what is the answer?
DNE or undefined
What should you do if the function is not provided? Ex. The data describes the charge Q remaining on the capacitator (in microcoulombs) at time t (s). Use the data to draw the graph of this function and estimate the slope of the tangent line at the point where t = 0.04 The following points are graphed as (t,Q): (0,100.00) (.02,81.87) (.04,67.03) (.06,54.88) (.08,44.93) (.1, 36.76)
Method 1 (table of values): average the slopes of the two secant lines before and after the time (0.04) ex. (63.03-81.87)/(.04-.02) = -742; (54.88-67.03)/(.06-.04) = -607.5 the average of these two is -674.75 = average estimated slope of the tangent line Method 2 (graph): draw a right triangle and find the slope (rise/run) The right triangle provided has points at (.06,80.4) and (.02,53.6); (80.4-53.6)/(.06-.02) = -670
Method 3: typing the function into y₁ and using the table
P(1,1) is constant and Q(x,x²) changes m(PQ) = (x² - 1)/(x-1) = slope of secant line at P and Q *CHANGE THE EQUATION FIRST - Select TABLESET and highlight ASK for the INDEPENDENT VARIABLE - Go to TABLE - Type in the x values and the slope will automatically be provided
Describe the symbolic relationship lim m(PQ) = m Q→P_________
Q→P means point Q gets closer to P mPQ is the equation for generic functions m is the slope of the tangent line
What mode should your calculator always be in for calculus?
RADIANS (calc) not DEGREES (physics)
Evaluate lim(x→∞) (x² + 3)/(x⁴ - 8) **highest degree in denominator
Shortcut: 0 Long: divide by highest degree in denominator (x² + 3)/(x⁴ - 8) * (1/x⁴)/(1/x⁴) = (1/x² - 3/x⁴)/(1-8/x⁴); sub in largest # = 0 + 0/ 1 - 0 = 0
Evaluate lim(x→∞) (3x² - x - 2)/(5x² + 4x + 1) **numerator and denominator have same power
Shortcut: numerator/denominator = 3/5 Long: divide each side by highest degree (3x² - x - 2)/(5x² + 4x + 1) * (1/x²)/(1/x²) = (3 - 1/x - 2/x²)/(5 + 4/x + 1/x²) sub in largest number you can think of for x; (3 - 0 - 0)/(5 + 0 + 0) = 3/5
Evaluate lim(x→-∞) (5x⁵ + 2)/(9x² - 2x + 1) **highest degree in numerator
Shortcut: ±∞ Long: Divide by highest degree in denominator (5x⁵ + 2)/(9x² - 2x + 1) * (1/x²)/(1/x²) (5x³ + 2/x²)/(9 - 2/x); sub in largest # = 5x³/9; what is the END behavior = -∞
Find lim(x→3⁺) 2x/(x - 3) and lim(x→3⁻) 2x/(x - 3)
Since f(x) is discontinuous at x = 3, check left and right limits: can be either ∞ or - ∞ RIGHT: pick number close to 3 on right ex. 3.5 numerator = +, denominator = + when substitute in; lim(x→3⁺) 2x/(x - 3) = ∞ LEFT: pick number close to 3 on left ex. 2.5 numerator = +, denominator = - when substitute in; lim(x→3⁻) 2x/(x - 3) = -∞
In order to find an equation of the tangent line to the parabola y = x² at the point P (1,1), what should you look for?
Since we know the point P, the only question is point m of the line
Show that there is a root in the given interval for the function f(x) = (x²-16)/(x-4); (1,5)
Step 1: Can I use IVT? No because it is not continuous within the interval; removable discontinuity at x = 4
Determine if there is a value in the given interval where f(c) = k if f(x) = -2x² - 4x + 3, k = -2, and interval is [-2,1]
Step 1: Can I use IVT? polynomial so continuous and in closed function; yes Step 2: Apply theorem f(-2) = 3; f(1) = -3; f(-2) < -2 < f(1) Step 3: State your determination Since f(x) is continuous on the closed interval [-2,1] and f(-2) < -2 < f(1) by IVT there exists a c value where -2 < c < 1 such that f(c) = -2
Show that f(c) = 7 in the given interval (-3,-1) for the function f(x) = x² + 2x + 3
Step 1: Can I used IVT? yes; continuous within a closed interval Step 2: Apply theorem f(-3) = 6; f(-1) = 2; 6 < 7 < 2 DNE ***can't apply theorem
How would you write the slope of a tangent line y - 3 = 5(x-2) at x = 1?
The value of the slope of the tangent line y - 3 = 5(x-2) at x =1 **always include as much information as possible and write it out completely
Use a table to find the limit of the following: lim √(t² + 9) - 3/t² t→0
Use TABLESET to plug values into your calculator and find the limit: -.1 = .16662 -.01 = .16667 -.001 = .16667 .001 = .16667 .01 = .16667 .1 = .16662 = 0.166. . . or 1/6
secant line
a curve that intersects a curve in at least two distinct points
How can you also consider continuity over an entire interval at a time?
a function f is continuous on an interval if it is continuous at every number on the interval (if f is defined only one one side of an endpoint on the interval, we understand continuous at the endpoint to mean continuous from the left or continuous from the right)
What does a function sometimes do as x approaches a number a from the left or from the right?
a function f(x) approaches two different limits
Why are all three graphs the same?
a limit describes a point the graph is approaching, not the hole itself; therefore a still approaches L even with a removable discontinuity or normal point
What are tangent lines?
a limit of secant lines
What is a tangent to a curve?
a line that touches the curve in one point
Where are each of the following functions discontinuous: a) f(x) = (x² - x - 2)/(x - 2) b) f(x) = { 1 if x = 0; { 1/x² if x ≠ 0 c) f(x) = { (x² - x - 2)/(x - 2) if x ≠ 2, 1 if x = 2 d) f(x) = ||x||
a) 2 (no 0 in denominator) b) 0 (point between 2 asymptotes, limit is infinite so DNE) c) 2 (# 3) d) greatest integer function so the limit DNE at all integers
Explain why each function is continuous or discontinuous a) the temperature at a specific location as a function of time b) the temperature at a specific time as a function of the distance due west from NYC c) the altitude above sea level as a function of the distance due west from NYC d) the cost of a taxi ride as a function of the distance traveled e) the current in the circuit for the lights in a room as a function of time
a) continuous; temp changes gradually b) continuous; temp changes gradually c) discontinuous; sea level changes abruptly d) discontinuous; changes at a different rate e) discontinuous; changes abruptly when lights turn on or off
What three things must be true for f to be considered continuous at a?
a) f(x) is defined (a is in the domain of f) b) lim(x→a) f(x) exists c) lim(x→a) f(x) = f(a)
Solve the following limits using algebraic calculations: a) lim(h→0) [(3+h)² - 9]/h b) lim(x→2) (x²-4)(x-2)
a) factor; lim(h→0) h(h+6)/h = lim(h→0) h + 6 = 0 + 6 = 6 b) lim(x→2) (x+2)(x-2)/(x-2) cancel; lim(x→2) x + 2 = 2 + 2 = 4
How can the limit of a function as x approaches a often be found?
direct substitution
Why doesn't the limit lim(x→0) 1/x² exist and what can this be written as?
doesn't exist because ∞ is not a number (infinite limit), but can be expressed as ∞
What should you ALWAYS DO ON TESTS (points taken off otherwise) if you put DNE?
explain why!!
What should you do when you have a tangent line?
find the slope
How do you find the value that the variable approaches for composite functions?
find the value for the first part; this is then inputted for x → __ for the limit of the second part of the function
When does lim f(x) = L ___________x→a_______
if and only if lim f(x) = L and lim f(x) = L _____________x→a⁻__________x→a⁺______
How can you think of a continuous function in an interval geometrically?
if it can be drawn without picking up your pen
When is a function considered continuous at a?
if lim f(x) = f(a) _x→a_________
How can f(a) be defined even when there is an hole?
if there is a disconnected point along the same axis of symmetry
When is the left-hand limit of f(x) as it approaches a (or the limit of f(x) as x approaches a from the left) equal to L for lim f(x) = L x→a⁻______
if we can make the values of f(x) arbitrarily close to L by taking x to be sufficiently close to a and x less than a
What is the slope of the tangent line equal to?
instantaneous velocity
How is f(x) = ||x|| an example of one sided continuity?
it is continuous from the right but discontinuous from the left
How is any polynomial continuous everywhere?
it is continuous on all real numbers (-∞,∞)
Is ∞ a number? What does this mean?
it is not a number; it can't be added, subtracted, multiplied, or divided
What must you know about a rational function to know where it is and is not continuous?
it's domain
What should you always make sure to do?
label units properly for real life applications
What is the limit for lim(x→a) c?
lim (x→a) c = c
How is the slope of the tangent line in relation to the slopes of the secant line expressed?
lim m(PQ) = m Q→P__________ OR lim x²-1/x-1 = 2 x→1____________
Prove that lim(x→0) |x|/x does not exist
lim(x→0⁺) |x|/x = lim(x→0⁺) x/x = lim(x→0⁺) 1 = 1 lim(x→0⁻) |x|/x = lim(x→0⁻) -x/x = lim(x→0⁻) (-1) = -1 right and left limits are different so DNE
Find lim(x→1) (x² - 1)/(x - 1)
lim(x→1) (x-1)(x+1)/(x-1) = lim(x→1) x + 1 1 + 1 = 2
What is the limit law for coefficients given that lim(x→a) f(x) = lim(x→a) g(x)?
lim(x→a) [cf(x)] = c lim(x→a) f(x) *the limit of a constant times a function is a constant times the limit of the function
What is the limit law for addition of functions given that lim(x→a) f(x) = lim(x→a) g(x)?
lim(x→a) [f(x) + g(x)] = lim(x→a) f(x) + lim(x→a) g(x) *the limit of a sum is the sum of the limits
What is the limit law for subtraction of functions given that lim(x→a) f(x) = lim(x→a) g(x)?
lim(x→a) [f(x) - g(x)] = lim(x→a) f(x) - lim(x→a) g(x) *the limit of a difference is the difference of the limits
What is the limit law for exponents?
lim(x→a) [f(x)]ⁿ = [lim(x→a) f(x)]ⁿ where n is a positive integer
What is the limit law for multiplication of functions given that lim(x→a) f(x) = lim(x→a) g(x)?
lim(x→a) [f(x)g(x)] lim(x→a) f(x) * lim(x→a) g(x) *the limit of the product is the product of the limits
According to the Squeeze Theorem (/Pinching Theorem/Sandwich Theorem), if f(x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a), what limits exist?
lim(x→a) f(x) = lim(x→a) h(x) = L then lim(x→a) g(x) = L
State the six statements where, if ANY of them are true, then there must be a vertical asymptote at x = a
lim(x→a) f(x) = ∞ lim(x→a⁺) f(x) = ∞ lim(x→a⁻) f(x) = ∞ lim(x→a) f(x) = -∞ lim(x→a⁺) f(x) = -∞ lim(x→a⁻) f(x) = -∞
What happens if f(x) ≤ g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a?
lim(x→a) f(x) ≤ lim(x→a) g(x)
What is the limit law for division of functions given that lim(x→a) f(x) = lim(x→a) g(x)?
lim(x→a) f(x)/g(x) = [lim(x→a) f(x)]/[lim(x→a) g(x)] IF lim(x→a) g(x) ≠ 0 *the limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0)
What is the limit for lim(x→a) x?
lim(x→a) x = a
What is the limit for lim(x→a) xⁿ?
lim(x→a) xⁿ = aⁿ where n is a positive integer
What is the limit for the square root of a function?
lim(x→a) ⁿ√f(x) = ⁿ√[lim(x→a) f(x)] where n is a positive integer assume that lim(x→a) f(x) > 0 if n is even
What is the limit for lim(x→a) ⁿ√x?
lim(x→a) ⁿ√x = ⁿ√a if n is even, assume that a > 0
Describe limits as x approaches +∞
look for the horizontal asymptote ex. y = x² - 1/x² + 1 = 1
The point P(8,2) lies on the curve y = √(x + 1) - 1. Write the equation of the tangent line to the curve at P(8,2).
mPQ = √(x + 1) - 3/(x - 8) plug into calculator as it approaches 8; = 1/6 y -2 = 1/6 (x - 8)
Can you use IVT under a function if a number is not provided?
no ex. cos x = x (0,1) but no value to test
Do you have to find the slope of the cosecant?
no, you can find the slope of the tangent by estimating the slope of the cosecant and inputting points for Q, but cannot find the slope of PQ because you do not know Q
Does lim(x→∞) cos x have a limit?
no; it is oscillating. DNE
If you can't apply IVT does it mean that the function will never equal N?
no; it just means that IVT cannot determine if it does
Do you have to put the equation of the tangent line in slope-intercept form from point-slope form?
no; point-slope form is preferred
What are the two ways you can find a limit?
numerically and graphically
What determines end behavior?
odd exponent = one side up one side down even exponent = both sides up or down negative coefficient = right side down positive coefficient = right side up
What do you need for the velocity equation?
only the slope (average or instantaneous), not the equation of the tangent line
Find lim(x→∞) (x² - x)
polynomial = end behavior; both ends positive facing up so = ∞
Show that there is a root in the given interval for the function f(x) = x⁴ + 2x³ - 4x - 3; (0,3)
root so try to find x-intercept (y = 0) Step 1: Can I use IVT? yes; continuous within a closed interval Step 2: Apply theorem f(0) = -3; f(3) = 120; f(0) < 0 < f(3) -3 < 0 < 120 Step 3: State your determination Since f(x) is continuous on the closed interval (0,3) and f(0) < 0 < f(3) by IVT there exists a c value where 0 < c < 3 such that f(c) = 0
The greatest integer function (step graph) is defined by [[x]] = the largest integer that is less than or equal to x. (For instance, [[4]] = 4, [[4.8]] = 4, [[π]] = 3, [[√2]] = 1, [[-1/2]] = -1.) Show that lim(x→3) [[x]] does not exist.
since [[x]] = 3 for 3 ≤ x < 4, lim(x→3⁺) [[x]] = lim(x→3⁺) 3 = 3 since [[x]] = 2 for 2 ≤ x < 3, we have lim(x→3⁻) [[x]] = lim(x→3⁻) 2 = 2 one-sided limits are not equal so lim(x→3) [[x]] = DNE
Method 1: Fill out the following table to approach 1 from one above and below: x | m(PQ) 2 | 1.5 | 1.1 | 1.01 | 1.001 | 0 | 0.5 | 0.9 | 0.99 | 0.999 |
square x values for Q then find slope | 3 ((4-1)/(2-1) = 3) | 2.5 | 2.1 | 2.01 | 2.001 | 1 | 0.5 | 0.9 | 0.99 | 0.999 **all values approach 2 so 2 is the slope
Find the limit for the following: lim(x→∞) (e⁻²ⁿ cos n)
squeeze theorem; [-1,1] lim -e⁻²ⁿ ≤ lim e²ⁿ cos n ≤ lim e²ⁿ
Intermediate Value Theorem (IVT)
suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a) ≠ f(b). Then there exists a number c in (a,b) such that f(c) = N
odd function
symmetrical about the origin; f(x) = (-x) obtain opposite of original function ex. y = 1/x, y = ³√x
even function
symmetrical about the y-axis; f(-x) = f(x) obtain original function ex. quadratic function, y = cos x
How should you check to see if a function is oscillating?
test graph and table for problems to see if it's oscillating and try to tell from the graph
oscillating (bouncing)
the closer you get to the origin, the more it switches but does not actually approach a specific number
What should you know?
the difference between finding limits as x approaches a and ∞
What is the equation for an object in free-fall?
the distance s(t) fallen after t seconds is given by s(t) = 4.9t²
What must a function be if it is continuous for IVT to work?
the function must hit every y value between f(a) and f(b) and hit N at least once
What is the inverse function of any continuous one-to-one function?
the inverse function is also continuous
What is the limit if x approaches infinity for a function?
the limit does not exist, but we use ∞ or -∞ to specify
What happens as you get close to the secant line?
you can find the tangent
What are the two possible answers for infinite limits?
∞ and -∞
What are the two options for a rational function with the highest degree in the numerator?
∞ or -∞, determined by end behavior no asymptote
If P is (1,1) and the parabola is represented by y = x², what are the coordinates of point Q?
(x,x²)
What should you find the slope between?
(1,1) and any ordered pair close to the nearby point Q(x,x²) on the parabola
Method 2: typing into the homescreen of your TI-84 If P(1,1) is a constant and Q(x,x²) is changing, how can this be represented in equation form and solved?
(f(x) -1)/(x -1) = (x² -1)/(x - 1) **always change the equation first Plug x values into the equation and solve ex. ((1.01)²-1)/(1.01 - 1) = 2.01 **can be simplified to x + 1 but rare case
State the domain for the following: - polynomials - rational functions - root functions - trigonometric functions - inverse trigonometric functions - exponential functions - logarithmic functions
- (-∞,∞) - all reals except when the denominator is 0 (ex. (-∞,3)u(3,∞)) - any x value for which the radicand is not 0 - cos/sin: (-∞,∞); tan: all reals except x ≠ (2n + 1)π/2 - sec: x ≠ (2n + 1)π/2; csc: x ≠ nπ; cot: x ≠ nπ - (-∞,∞) - (0,∞)
Solve the following: - lim(θ→0) cos θ - lim(θ→0) sin θ
- 1; (0,1) is an ordered pair so as it approaches 0 approaches 1 - 0; (0,0) is an ordered pair for the graph of sin x so as it approaches 0 it approaches 0 (the origin)
Consider: the piecewise graph y = g(x) with a floating point at (5,1) and removable discontinuities at (2,3) which goes to the left, and (2,1) which goes to the right and includes the hole (5,2). Find the limits of the following: - lim (x→2⁻) g(x) - lim (x→2⁺) g(x) - lim (x→2) g(x) - lim (x→5⁻) g(x) - lim (x→5⁺) g(x) - lim (x→5) g(x)
- 3 - 1 - DNE (3≠1) - 2 - 2 - 2 (same thing; approach same point)
What functions are also continuous at a if f and g are continuous at a and and c is a constant?
- f + g - f - g - cf - fg - f/g
Describe one-sided continuity: When is the function f continuous from the right at number a? When is the function f continuous from the left at number a?
- lim(x→a⁺) f(x) = f(a) - lim(x→a⁻) f(x) = f(a)
What must apply to the function for you to be able to use the IVT?
- must be CONTINUOUS on a CLOSED INTERVAL - must be BETWEEN f(a) and f(b)
What two cases for limits use DNE?
- oscillating graphs - functions where the left and right side of the graph approach different numbers
What functions are continuous at every number in their domains?
- polynomials - rational functions - root functions - trigonometric functions - inverse trigonometric functions - exponential functions - logarithmic functions
State the limit for the following graphs given that lim(x→a) f(x) = L in all three cases: - a graph with the coordinate (a,L). There are no removable discontinuities - the same graph with a removable discontinuity at the point (a,L) and a disconnected point below it along x = a (a, f(a)) - the same graph as the first with a removable discontinuity at the point (a,L)
- the limit is L - the limit is L - the limit is L as the function approaches a, it approaches L
Describe how each requirement could not be met: - f(a) is defined (in domain of x) - lim(x→a) f(x) exists - lim(x→a) f(x) = f(a)
- there is a removable discontinuity/hole or asymptote - piecewise function (different values when you approach the limit from the left or right) - you plug in a number but the answer is a different number
Given the following descriptions of places where a graph is discontinuous, describe the type of discontinuity and the reason why it is discontinuous using the definition: 1) at x = 1, a removable discontinuity is in the middle of an uninterrupted line 2) at x = 3, a real point exists below the x axis and a removable discontinuity on the same a.o.s. above the x axis 3) at x = 5, a removable discontinuity is in the middle of an uninterrupted line with a floating point above it
1) type: removable discontinuity; reason: f(1) is undefined 2) type: jump discontinuity; reason: lim(x→3⁻) f(x) ≠ lim(x→3⁺) f(x) 3) type: removable discontinuity (even w/ pt); reason: lim(x→5) f(x) ≠ f(5)
Find lim(x→0) x² sin 1/x
1. rewrite as an inequality using the range of the function range of sin = [-1,1] -1 ≤ sin 1/x ≤ 1 2. multiply by x² -x² ≤ sin x²/x ≤ x² 3. take the limit of the left, middle, and right as x→0 lim(x→0) -x²≤ lim(x→0) sin x²/x ≤ lim(x→0) x² 0 ≤ x ≤ 0 so lim(x→0) x² sin 1/x = 0
Given a graph where f(x) is a limit with a hole at (-3,1) and points at (1,2) and (3,0) and g(x) is a piecewise function where the left side includes the point (-2,-1) and ends at (1,-2) and the right side includes a hole at (-1,1) and points at (2,0), (3,1), and (4,2), use the limit laws to evaluate the following limits: a) lim (x→-2) [f(x) + 5g(x)] b) lim(x→1) [f(x)g(x)] c) lim(x→2) f(x)/g(x)
a) lim(x→-2) f(x) = 1 and lim(x→-2) = -1 lim(x→-2) [f(x) + 5g(x)] = (1 + 5(-1)) = -4 b) lim(x→1) f(x) = 2 but lim(x→1) g(x) *does not exist* because the left and right limits are different: lim(x→1⁻) g(x) = -2 and lim(x→1⁺) g(x) = -2 = DNE c) lim(x→2) f(x) ≈ 1.4 and lim(x→2) g(x) = 0 g ≠ 0 because it is in the denominator so = DNE
Describe infinite limits at infinity
a) lim(x→-∞) f(x) = ∞ b) lim(x→∞) f(x) = -∞ c) lim(x→-∞) f(x) = -∞ either shooting up or shooting down **no limit laws bc it is not a #
Evaluate the following limits and justify each step: a) lim(x→5) (2x² - 3x + 4) b) lim(→-2) (x³ + 2x² - 1)/(5 - 3x)
a) lim(x→5) (2x²) - lim(x→5) (3x) + lim(x→5) 4 = 2 lim(x→5) x² - 3 lim(x→5) x + lim(x→5) 4 2(5²) - 3(5) + 4 = 39 b) lim(x→-2) (x³ + 2x² - 1)/ lim(x→-2) (5 - 3x) [lim(x→-2) x³ + 2 lim(x→-2) x² - lim(x→-2) 1)]/[lim(x→-2) 5 - 3 lim(x→-2) x] [(-2)³ + 2(-2)² - 1]/[5 - 3(-2)] = -1/11
How should you graph trig functions?
after inputting them into Y =, select ZOOMTRIG
What does the definition of a continuous function implicitly require?
all three things to be true for f to be continuous at a
What should you do with composite functions?
always work from the inside out
What is Q?
any point generated on the x axis
How should you express values?
at what time (ex. t = 1 s) and with units ex. instantaneous velocity = 5 m/s at t = 5
What is the slope of the secant line equal to?
average velocity
Complete the following graph for instantaneous velocity after 5 seconds given that s(t) = 4.9t²: time interval: 5 ≤ t ≤ 6 5 ≤ t ≤ 5.1 5 ≤ t ≤ 5.05 5 ≤ t ≤ 5.01 5 ≤ 5 ≤ 5.001
average velocity: 53.9 49.49 49.245 49.049 49.0049 Looking at shorter intervals the instantaneous velocity appears to be 49 m/s at 5 seconds
****Find the limit of the function given the graph below: lim (sin x)/x x→0_______ ___________|________ ___(1)__ __-- __ ____ ____--_____|____--__
based on the graph, the limit is 1; important function to remember for AP test
What is the formula for average velocity?
change in position/time elapsed
How can you approximate the desired quantity?
computing the average velocity over a brief period of time such as a tenth of a second
Explain why the following function is continuous at every number in its domain and state the domain: f(x) = sin⁻¹(x² - 1)
continuous because inverse trig function; domain is [-1,1] bc range of sin -1 ≤ x² - 1 ≤ 1 [-√2, √2]
Consider the function f defined by f(x) = x² -x + 2 for values of x near 2. Visualize a parabola and the coordinate (2,4); the line the coordinate is on approaches it from BOTH directions. What is the limit for 2 and how would you express this?
the limit for 2 is 4 because it is the y value. You would express this as: lim (x² - x + 2) = 4 x → 2_____________ Basically, the limit of x² - x + 2 as it approaches 2 is 4
What is the slope of the tangent line?
the limit of all the slopes of the secant lines
How would you describe the following? lim f(x) = L x→a______
the limit of the function f(x) as it approaches a is L
If a ball is dropped from the top of a tower 450 meters tall and you want to find the velocity after 5 seconds, what is the instantaneous velocity for the equation?
the limiting value of the average velocities over shorter and shorter time periods that start at t = 5
horizontal asymptote
the line y = b is a horizontal asymptote because of the graph of a function y = f(x) if either lim(x→∞) f(x) = b or lim(x→-∞) f(x) = b
What should you use P and Q to find?
the slope m(PQ) of the secant line PQ
What is the x value for when the graph hits N (y value)? What happens if there is more than one point at which it hits N?
the x value is c; you can use c1, c2, c3, etc.
What happens if one of these three requirements is not met?
then f is discontinuous at a
How has continuity at an interval been defined so far?
to occur one point at a time
What kind of functions do you use the squeeze theorem with?
trig functions ex. ranges sin = [-1,1] cos = [-1,1] tan = (-∞,∞)
What should you do if you have an normal number outside of radical?
use 1/x for the normal number and 1/√x² for the radicand
Investigate lim sin (π/x) x→0________ using a table and a graph
values in the table are 0 for every value surrounding 0; graph shows lines that get closer and closer together and bounce as they move towards the origin = oscillating so DNE (does not exist)
What happens as the time approaches 5 seconds?
velocity approaches average velocity ex. average velocity for 5.1 s(5.1) - s(5)/(5.1-5.0) 4.9(5.1)² - 4.9(5)²/(5.1-5.0) = 49.49 m/s
When does a removable discontinuity (hole) exist?
when a factor in the numerator cancels with a factor in the denominator
When is any rational function continuous?
wherever it is defined, wherever it is continuous on its domain
When do you use the Squeeze Theorem?
with trigonometric functions that are multiplied or divided
Locate the discontinuities of the function y = 1/(1 + e¹/ⁿ) (assume ⁿ = x)
x = 0 because exponents cannot have 0 in the denominator; undefined
Show that lim(x→0) |x| = 0
x = {x if x ≥ 0, {-x if x < 0 (one should be closed not both) for |x| = x for x > 0 lim(x→0⁺) |x| = lim(x→0⁺) x = 0 for x < 0 |x| = -x so lim(x→0⁻) |x| = lim(x→0⁻) (-x) = 0 therefore lim(x→0) |x| = 0 **use for both sides of y = |x| (left and right)
What should x not be equal to and why?
x ≠ 1 so Q ≠ P **start one below and one above, 0 and 2
Find the equation of the desired tangent line given the point (1,1) and that the slope is 2
y - 1 = 2(x - 1) OR y = 2x - 1