April 18, 2017 REVIEW FOR AP CALCULUS EXAM Review: Limits A. what is a limit? y-value being approached as the function approaches a certain x-value Doesn't matter whether or not the value is but actually reached; just that the point is being approached from both sides
B. types of discontinuities hole jump asymptote
at a hole, the limit exists at a jump, the limit D.N.E. at a vertical asymptote, the limit D.N.E. (or is ±∞)
B. definition of continuity at a point Example of a discontinuity where each 1. f(c) exists statement is not true: value at x = c exists lim 2. x - > c f(x) exists limit as x -> c exists
lim 3. f(x) = f(c) x->c limit as x->c = value when x = c
D. finding limits algebraically In this case, -continuous functions plug in x = c and
continuous because no domain restrictions (no dividing by 0 or square roots of negatives) -functions with discontinuities If you get 0/0 when plugging in: The limit may exist. Factor out the discontinuity, cancel, and then plug in x = c.
cancel the factor causing the 0/0 (hole), and then If you get (a non-zero #)/0: plug in x = 2 to the continuous function The limit does not exist (vertical asymptote). April 18, 2017 -piecewise-defined functions
If If
find find a value of k that makes f a continuous Two ways to do this: function (at x = 2). 1. Graph the function and see if the two Plug in x = 2 at the endpoint, and make sides are approaching the same value the y-values equal to each other
2. (easier) see if the endpoints are the same by plugging in x = 2
Another method for finding limits: L'Hospital's Rule
In other words, if plugging in the value gives 0/0 (or ±∞/∞), take the derivative of the top and bottom, and plug in again. Example 1: Example 2: April 18, 2017
E. finding limits as x → ±∞ (horizontal asymptotes, aka end behavior)
if (deg. of top) = (deg of bottom)
if (deg. of top) < (deg of bottom)
if (deg. of top) > (deg of bottom)
other function types
Review: Rules of Derivatives 1. What is the meaning of a derivative? (How could you approximate it from a graph or table?) derivative means instantaneous slope, or slope at a point
2. continuous vs. differentiable continuous: no "holes"/jumps/asymptotes/endpoints differentiable: slope is continuous at that point (not vertical, not a sudden change in slope) A function can be continuous but not differentiable.
It cannot be differentiable but not continuous. April 18, 2017
F. one-sided limits Example:
+
If = , then exists. +
G. Intermediate Value Theorem If f is continuous on [a,b] and f(a) ≠ f(b), then for all c with f(c) between f(a) and f(b), f(c) exists on (a,b).
H. Extreme Value Theorem If f is continuous on [a,b], then f has an absolute/global maximum and an absolute/global minimum on that interval, which occur either at endpoints or at critical numbers.
If the function is not continuous, this does not have to be true:
3. What is the definition of a derivative? How do we use it to find the derivative of a function, like y = x2? A derivative is the limit of the slopes between 2 points as the second point gets "infinitely close" to the first point. Three forms:
In this case, we're not actually going to take a derivative. We need to notice that this is basically just a definition of a derivative (same setup as the first equation above.) So the whole thing equals f '(x). The function f is sin, and instead of x, we're using π/3. So we're finding the derivative of sin, and then plugging in x = π/3. April 18, 2017 4. average rate of change (for a function from x=a to x=b) vs. instantaneous rate of change (at x=c)
slope of secant line:
slope of tangent line: slope at a point (derivative)
5. Mean Value Theorem for derivatives If f is continuous and differentiable on [a,b], then for some x = c on the interval, f(b) - f(a) f '(c) = b - a Meaning: If f is continuous and differentiable, the function's slope must equal its average slope somewhere on an interval.
6. Rules of derivatives E. Exponential functions and logs A. power rule Bring down the power, and then subtract 1 from the power.
B. product rule F. trig functions
D. chain rule April 18, 2017 G. implicit differentiation: When do we use it, and how? used when equation is not solved for y (written "implicitly") -- derivative of each term involving y is multiplied by dy/dx (deriv of inside function) Example 1: Differentiate y2 - 3xy +x2 = 1
Factor out y ', and then divide by the rest of that side
Be able to find second derivatives implicitly as well: Example 2: If x2 - 2y2 = 14, find the value of d2y / dx2 at the point (4,1). April 18, 2017 H. Derivatives of inverse functions (NEW!!) If two functions are inverses, their derivatives are reciprocals (at the switched point).
Example: If f(2) = 3, f '(2) = 4, f(3) = 6, and f '(3) = -2, and if g(x) = f-1(x), find g '(3).
I. Derivatives of inverse trig functions (NEW!!)
Example: If y = arcsinx, find dy/dx.
Shortcuts: y y' y
arcsinx arccosx
arctanx arccotx
arcsecx arccscx April 18, 2017
Applications of Derivatives 1. slope of a tangent line
2. equation of a tangent line y = m(x - h) + k
3. tangent lines given information Example 1: Find the equation of the line tangent to the function f(x) = 4sinx + 1 when x = π/2.
Example 2: Find the line tangent to the curve f(x) = x4 - 2x2 - 6x when f '(x) = 4.
4. critical numbers and local extrema Critical numbers are x-coordinates at which the derivative is 0 or undefined. These points are the only possible local extrema (but may not all be local extrema).
5. increasing/decreasing functions Example: On what interval(s) is the function f(x) = x3 - 2x increasing? April 18, 2017
6. graphs of functions and derivatives
f(x) f '(x) f ''(x)
has positive values increasing (above x-axis) has negative values decreasing below x-axis) (is 0 or undefined at a local max and) changes from positive to negative (is 0 or undefined at a local min and) changes from negative to positive accelerating -- has positive values CONCAVE UP (above x-axis) decelerating -- has negative values CONCAVE DOWN below x-axis)
Examples: Graphs of derivatives f f '
f ' f April 18, 2017 7. first derivative test If the derivative of f is changing from positive to negative at a critical number, then f has a local ______max at that critical number.
If the derivative of f is changing from negative to positive at a critical number, then f has a local ______min at that critical number.
8. absolute extrema Absolute extrema are the overall highest and lowest points on a function. These points occur either at relative extrema or at endpoints. To find absolute extrema: Draw a sign chart, and look for relative maxes and mins. Then consider the endpoints. Identify all possible locations for absolute max/min, and plug in to find the y-values. Whichever one is largest/smallest is the max/min.
9. optimizing functions Basically, you're finding the absolute max/min. 1. write equation to be optimized 2. find the derivative, and find the critical numbers (derivative = 0 or undefined) 3. identify the critical number (or endpoint) that gives a max or min 4. find the max or min value (y-value)
Example: If y = 4x - 6, find the minimum value of xy. Equation to be minimized: M = xy Too many variables, so replace y by what it equals (in terms of x):
M = x(4x - 6)
Before taking the deriv, rewrite to avoid the product rule:
M = 4x2 - 6x
Now take the derivative and set = 0 to find the critical numbers, and then draw a sign chart: Plug surrounding values into M' (this is a M' = 8x - 6 = 0 derivative sign chart), like x = 0 and x = 1. 8x = 6 x = 3/4
So the absolute min occurs when x = 3/4. Last step: FIND the absolute min (y-value).
M(3/4) = (3/4)(4 • 3/4 - 6) = (3/4)(3 - 6) = (3/4)(-3) = -9/4 April 18, 2017
10. related rates Related rates are used to compare two rates of change (look for changes with respect to time). Each variable is really a function of t, so each derivative is multiplied by d*/dt.
Example: If the area of a circle is increasing at 2π m/s, find the rate at which the circumference of the circle is increasing when r = 6 m. This is a particularly tricky question because area and circumference cannot be (easily) written in the same equation. Start with what you know, and figure out what you can:
rate of change in area = dA/dt = 2π m2/s radius = 6m dC/dt = ? (what we're looking for) Area of a circle: A = πr2 Circumference of a circle: C = 2πr
We can use the area equation: A = πr2 dA/dt = 2πr • dr/dt Plugging in values can solve for dr/dt: 2π = 2π(6) • dr/dt (2π) / (12π) = dr/dt dr/dt = 1/6 Now let's try to find dC/dt: C = 2πr dC/dt = 2π • dr/dt dC/dt = 2π • (1/6) dC/dt = 2π/6 = π/3 m/s
11. position, velocity, acceleration If f is position, then f ' is velocity (rate of change), and f '' is acceleration (rate of change in velocity). f = position
f ' = velocity 12. concavity f " = acceleration If f '' is positive, then f is concave up.
If f '' is negative, then f is concave down.
13. inflection points Inflection points are x-coordinates at which f '' changes signs. (Points where f '' is zero or undefined only count as inflection points if f '' changes signs.) Note: Inflection points are the second derivative version of critical numbers (where f ' is 0 or undefined). HOWEVER, the points where f " = 0 or undefined but doesn't change signs don't count in inflection points. The second derivative must actually change signs. April 18, 2017
14. second derivative test If f '' is positive at a critical number, then f has a local minimum at that critical number.
If f '' is negative at a critical number, then f has a local maximum at that critical number. Idea: Critical numbers are points where the derivative can change sign, so the function can change direction.
If the concavity is negative at one of these points, then it could look like this: (local max)
If the concavity is positive at one of these points, then it could look like this:
(local min)
Rules of Integration 1. What is an integral? area between the curve and the x-axis on the given interval
definition:
2. properties of integrals April 18, 2017 3. Fundamental Theorem of Calculus or To find the area underneath a curve, find any antiderivative, and subtract top - bottom.
4. 2nd form of the FTC The derivative of an integral is the original function.
5. indefinite vs. definite integrals indefinite: no limits of integration - find the general antiderivative (+C needed)
definite: limits are given, so find the area (no C needed)
6. undoing the power rule April 18, 2017
7. u-substitution used to undo the chain rule! Process: Look for a function inside another function, or a function whose derivative is also in the integral.
Example:
8. Mean Value Theorem of Integrals (average value) To find the average value/average height of a function on an interval, divide the function's area by the width of the interval.
Example: Find the average acceleration of a function whose acceleration is defined by a(t) = 4t3 - 5t + 2. April 18, 2017
9. accumulation functions (integrating rate functions) The integral of a rate function gives the total change in value, or total accumulation, over a time period.
Example: If a cake is taken from a 350º oven and cools at a rate of r(t) = 100e-.9t, what will its temperature be after 10 minutes?
10. volumes R = function (if revolving -disc method NO HOLE around x-axis) Used when a region is revolved around an axis perpendicular to its representative rectangle (i.e. horizontal line).
-washer method HOLE Same as above, but the revolved region contains a hole on the inside.
r = function closer to axis of revolution R = function further What if rotating around a line other than an axis? from axis of revolution Each radius is (line - function), rather than just the function.
When do we use y's in the integral? Use y when a region is revolved around a vertical line.
-cross-sections Find the area of the cross-section, and integrate.
Base of the cross-section is ALWAYS (top - bottom)
Don't include π unless it's part of the area formula April 18, 2017 Example (2013 AP, Question 5 - no calculator): Let f(x) = 2x2 - 6x + 4 and g(x) = 4cos((1/4)πx). Let R be the region bounded by the graphs of f and g, as shown in the figure. a. Find the area of R.
b. Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the horizontal line y = 4.
c. The region R is the base of a solid. For this solid, each cross-section perpendicular to the x-axis is a square. Write, but do not evaluate, an integral expression that gives the volume of the solid.
11. solving separable differential equations Separate the y's and dy's from x's and dx's. Then integrate, solve for y, and (if necessary) solve for C.
12. dy/dt = ky exponential growth (change in y is some multiple of the current y-value)
13. slope fields gives a graph of all possible antiderivatives by plotting values of the derivative at any point (slopes of f) To identify a slope field, first find points where the slope should be 0.