Combined Chapters
Displacement
Integral of v(t)
Finding volume given equilateral triangle with sides of length x as cross sections
( √3/4)x^2
Suppose you drive 200 miles, and it takes you 4 hours, what is your average speed?
(200-0)/(4-0) = 50 mi/hr
Average value of a function
(the integral from b to a of f(x) dx) / (b-a)
Volumes of Solids of Revolution: The Disc Method
1. Always rotate around an EDGE of the region; it must touch the region. 2. "Slice" or rectangle is perpendicular to axis or revolution creating a cylinder. v = π∫(r^2) dx
How to calculate rectangular approximation
1. Divide interval [a,b] into subintervals 2. Create rectangles using width of each subinterval as the base 3. Finding the height using one of the three following options: a. use left endpoint of each interval (LRAM) b. use right endpoint of each interval (RRAM) c. use the midpoint of each interval (MRAM)
Definite Integrals with substitution
1. Leave the limits in terms of the of the original variable and integrate like you did for the indefinite integrals. Once you have returned all variables back to the original letter, you can plug in the upper and lower limits. 2. Using the rule for the change of variables, change the limits with the same rule, then you never need to return to the original. **The limits must match the variable being used, or there must be some notation to indicate that the limits being used are different from the variable being
U-Substitution
1. Let the "inside" function be u. 2. Find du/dx 3. Substitute the inside function with u and multiply by du/dx 4. Differentiate with respect to u. 5. Substitute u back in.
Strategy for Solving Related Rate Problems
1. Understand the problem. • In particular, identify the variable whose rate of change you seek and the variable (or variables) whose rate of change you know. 2. Develop a mathematical model of the problem. • Draw a picture (many of these problems involve geometric figures) and label the parts that are important to the problem. Be sure to distinguish constant quantities from variables that change over time. Only constant quantities can be assigned numerical values at the start. 3. Write an equation relating the variable whose rate of change you seek with the variable(s) whose rate of change you know. • The formula is often geometric, but it could come from a scientific application. 4. Differentiate both sides of the equation implicitly with respect to time t. Be sure to follow all the differentiation rules. The Chain Rule will be especially critical, as you will be differentiating with respect to the parameter t. 5. Substitute values for any quantities that depend on time. • Notice that it is only safe to do this after the differentiation step. Substituting too soon, "freezes the picture" and makes changeable variables behave like constants, with zero derivatives. 6. Interpret the solutions. • Translate your mathematical result into the problem setting (with appropriate units) and decide whether the result makes sense.
Functions are continuous at c if
1. f(c) is defined 2. lim x →c f(x) exists 3. lim x →c f(x) = f(x)
How to find the particular solution to a differential equation given initial conditions
1. separation of variables 2. integrate both sides of equation 3. plug in initial condition to find "c" 4. plug "c" into equation in step 3. 5. isolate y
Differential Equations
A differential equation is an equation that relates a function (or relation) with its derivatives. Just like in Algebra, when you want to solve an equation, you use an inverse operation. To "undo' a derivative we find antiderivate.
Theorem 3
A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal.
How f'(a) Might Fail to Exist
A function will not have a derivative at a point P(a, f(a)) where the slopes of the secant lines, f(x)-f(a)/(x-a) fail to approach a limit as x approaches a.
One-Sided Derivatives
A function y = f(x) is differentiable on a closed interval [a,b] if it has a derivative at every interior point of the interval, and if the following limits exists at the endpoints
Properties of Limits as x→ ±∞ Limits at infinity have the same properties as those of infinite limits. Review the previous section for properties.
A general rule for functions that are divided is • If the denominator grows faster than the numerator, the limit as x approaches infinity will be 0. • If the numerator grows faster than the denominator, then as x approaches grows infinity, the limit will increase or decrease without bound. • If the numerator and denominator grow at the same rate, look for the a limit of a non-zero constant.
Related Rates Problem
A hot-air balloon rising straight up from a level field is tracked by a range finder 500 feet from the lift-off point. At the moment the range finder's elevation angle is pi/4, the angle is increasing at the rate of 0.14 radians per minute. How fast is the balloon rising at that moment?
Increasing & Decreasing Functions
A positive derivative means that the function is increasing. A negative derivative means that the function is decreasing. A zero derivative means that the function has some special behavior at the given point. It may have a local maximum or a local minimum.
Area of a trapezoid
A=1/2h(b1+b2)
Finding critical numbers
All local extrema occur at critical points of a function — that's where the derivative is zero or undefined (but don't forget that critical points aren't always local extrema; there must be a sign change). The first step in finding a function's local extrema is to find its critical numbers (the x-values of the critical points)
Finding inflection points
An inflection point is where a curve changes concavity, which can be found by setting f''x = 0 and looking for sign change.
U-Substitution
An integration method that essentially involves using the chain rule in reverse. 1. Let u be the "inside function" and du =u'(x)dx. 2. Find ∫udu 3. Re-substitute so expression in the terms of the original function.
Related Rates
Any equation involving two or more variables that are differentiable functions of time t can be used to find an equation that relates their corresponding rates. These types of problems are called Related Rates. When one or more values in an equation change over time, we have related rates. We use related rates when the problem asks: How fast did something change? If h is measured in cm and t is measured in minutes, then dh/dt is measured in: cm/min
Riemann Sum
Approximating Area under the curve using rectangles
The Quotient Rule
At a point where g(x) ≠ 0, the quotient y = f(x)/g(x) of two differentiable functions is differentiable, and
Concavity of graphs/functions
Concave up: f''x>0 Concave down: f''x<0
Methods of evaluating limits: graphical analysis
Does the appear to approach a y-value as x approaches the limit?
Methods of evaluating limits: numerical analysis
Evaluating the function at x-values very close to the limit on both sides.
The Chain Rule
Formal Definition: If f is differentiable at the point u = g(x), and g is differentiable at x, then the composite function (f ∘ g)(x) = f(g(x)) is differentiable at x, and
Intermediate Value Theorem (IVT)
Functions that are continuous on intervals have properties that make them particularly useful in mathematics and its applications. A function is said to have the intermediate value property if it never takes on two values without taking on all the values in between
Logistic Growth
Growth pattern in which a population's growth rate slows or stops following a period of exponential growth
Exponential Growth
Growth pattern in which the individuals in a population reproduce at a constant rate.
Implicitly Defined Functions
How do we find the slope when we cannot conveniently solve the equation to find the functions. For example the function x^3 + y^3 - 9xy = 0 ? We treat y as a differentiable function of x and differentiate both sides of the equation with respect to x, using the differentiation rules for sums, products, quotients, and the Chain Rule. Then solve dy/dx for in terms of x and y together to obtain a formula that calculates the slope at any point (x, y) on the graph from the values of x and y. This process is called implicit differentiation.
Tangent Line Approximation
If a function f is differentiable at x=a, then use can use the equation L(x) = f(a) + f'(a) (x-a) or y-y1=m(x-x1) to find the approximate tangent line at that point.
Differentiability Implies Continuity
If f has a derivative at x = a, then f is continuous at x = a. (the converse is not true)
Mean Value Theorem for Integrals
If f is continuous on [a,b], then there exists a c in [a,b] such that integral from a to b of f(x) dx = f(c)(b-a).
Second Fundamental Theorem of Calculus
If f is continuous on an open interval containing a, then for every x in the interval the derivative of the the integral of f(x) dx on said interval is equal to f(x)
Derivative of a Constant Function
If f is the function with the constant value c, then,
Power Rule for Integer Powers of x
If n is a positive or negative integer, then,
Power Rule
If r and s are integers, s ≠0, then lim (f(x))^(r/s) = L^(r/s)
Infinite Limits as x→ a
If the values of a function f(x) outgrow all positive bounds as x approaches a finite number a, we say that lim as x→ a f(X) = + ∞. Similarly, if the values of f become large and negative, exceeding all negative bounds as x approaches a, we say that lim as x→ a f(x) = -∞
The Sum and Difference Rule
If u and v are differentiable functions of x, then their sum and difference are differentiable at every point where u and v are differentiable. At such points,
The Constant Multiple Rule
If u is a differentiable function of x and c is a constant, then,
How to use/write IVT on a FRQ
If y is between f(a) and f(b), then y = f(c) for some c in [a,b]
IVT Example
If you are 5ft on your 13th birthday and on your 14th birthday you are 5'6", then at some point you had to be 5'4"
Distance
Integral of absolute value of v(t) *distance cannot be negative
Continuity
Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function.
rectilinear motion
Movement that occurs in a straight line.
Slope field
Plug (x,y) coordinates into differential equation, draw short segments representing slope at each point
Slope of a Curve y = f(x) at a Point P(a, f(a))
Provided the limit exists. What you are taking the limit of above is called the difference M= li ______________ ____________. If you are asked to find the slope using the definition or quotient using the difference quotient, this is the technique you will use.
Right-hand and left-hand limits
Sometimes the values of a function f tend to different limits as x approaches a number c from opposite sides. When this happens, we call the limit of f as x approaches c from the right the right hand limit of f at c and the limit as x approaches c from the left the left hand limit of f at c.
Mean Value Theorem
States that if a function f is continuous on [a,b] and differentiable on (a,b) then there exists a real number c on the interval such that:
End Behavior Model
The end behavior of a polynomial function is the behavior of the graph of f(x) as x approaches positive infinity or negative infinity. The degree and the leading coefficient of a polynomial function determine the end behavior of the graph.
Constant Multiple Rule
The limit of a constant times a function is the constant times the limit of the functions
Product Rule
The limit of a product of two function is the product of their limits
Quotient Rule
The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero
Difference Rule
The limit of the difference of two functions is the difference of their limits
Sum Rule
The limit of the sum of two function is the sum of their limits
vertical asymptotes
The line x = a is a vertical asymptote of the graph of a function y = f(x) if either lim x→ a+ f(x) = ±∞ or lim x→ a- f(x) = ±∞
Horizontal Asymptote
The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either lim as x→ +∞ = b or lim as x→ -∞ = b
Normal to a curve
The normal line to a curve at a point is the line perpendicular to the tangent at that point
The Product Rule
The product of two differentiable functions u and v is differentiable, and
Instantaneous Rate of Change
The speedometer in your car does not measure average velocity, but instantaneous velocity. The velocity at one moment in time
derivative of inverse function
To find the derivative of the inverse at the point (a, b) we find the reciprocal of the derivative of f as the point (b, a)
Derivative of a Composite Function
We now know how to differentiate sinx and 3x + 1, but how do we differentiate a composite like sin(3x + 1)? The answer is the Chain Rule, which is probably the most widely used differentiation rule in mathematics.
Finite Limits as x→ ±∞
We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. When we say, "the limit of f as x approaches infinity" we mean the limit of f as x moves increasingly far to the right on a number line. When we say, "the limit of f as x approaches negative infinity" we mean the limit of f as x moves increasingly far to the left.
Washer method
We will call the outer radius R, and we will call the inner radius r. The height of the cylinder formed is just the width of the strip. Just like before, if we have an infinitely thin strip, this distance will be denoted dx (if it is a vertical strip) and dy (if it is a horizontal strip)
Definition of Derivative
When it exists, this limit is called the derivative of f at a. The derivative of the function f with respect to the variable x is the function f' whose values at x is f'(x) = lim h-->0 (f(x+h)-f(x))/h , provided the limit exists. The domain of f' may be smaller than the domain of f. If f'(x) exists, we say that f has a derivative (is differentiable) at x. A function that is differentiable at every point of its domain is a differentiable function
Acceleration (instantaneous)
a(t) describes the rate of change in the velocity. a(t) = v'(t) = s"(t) • We can use the velocity and acceleration to determine the following: o An object is speeding up when v(t) and a(t) have the same sign. o An object is slowing down when v(t) and a(t) have different signs. different
Methods of evaluating limits: algebraically
a. Direct Substitution b. If Direct substitution yields an indeterminate form (0/0), then use some algebraic technique to simplify
Derivative of a^x
a^x ln(a)
optimization
calculating the minimum or maximum value of a function. For example, companies often want to minimize production costs or maximize revenue
If the function is __________________, THEN the Definite Integral will exist.. However, the converse, while true some of the time is NOT ALWAYS TRUE
continuous
The derivative of f(x)
d/dx(f(x))
The derivative of f with respect to x
df/dx
The derivative of y with respect to x
dy/dx
The derivative of f with respect to x
f'(x)
How to use your TI calculator to find a Definite Integral
fnInt(function, x, lower bound, upper bound) 1. Press MATH 2. Press 9: fnInt( 3. Follow the syntax above (be sure to enter comma between each)
Recall, that a function can have many antiderivatives, all of which vary by a constant. The ___________ of a differential equation is the order of the highest derivative involved in the equation.
order
𝑠(𝑡) = −16𝑡^2 + h
position function if an object is dropped from an initial height h
Position
s(t) describes the position s of an object after t seconds. • Displacement of an object is the change in position over a given interval of time. • On a position graph, we can easily determine the direction an object moves: - When s(t) is increasing, an object moves up or right. - When s(t) is decreasing, an object moves down or left.
absolute extrema (maximum/minimum)
the absolute maximum and minimum are the highest and lowest y-values on a given interval. 0) Verify that the function is continuous on the interval [a,b] 1)Find all critical points of f(x) that are in the interval [a,b] by setting f'(x)=0 2)Evaluate the function at the critical points found in step 1 and the end points. Make a chart to show the x and f(x) values in an organized manner 3) Identify the absolute extrema.
Average Rate of Change
the amount of change divided by the time it takes. In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval.
instantaneous speed
the speed of an object at one instant of time
average velocity
the total displacement divided by the time interval during which the displacement occurred. Average velocity, however, can be a negative value when the distanced covered is negative (ex: an object falling from a cliff).
relative maximum/minimum
the y-coordinate of any point that is the highest/lowest point for some section of the graph 1) find the derivative of f(x) 2) Finding all critical points and all points where f'(x) is undefined or equal to zero. 3) Let f be continuous on an interval I and differentiable on the interior of I. If f'(x)> 0 on the interval, then f is increasing on I. If f'(x)<0 on the interval, then f is decreasing on I
average speed
total distance divided by total time. Speed is always positive because the distance covered is positive as is the elapsed time.
Trapezoidal Rule
use trapezoids to evaluate integrals (estimate area)
Velocity (instantaneous)
v(t) describes the rate of change in the position. v(t) = s'(t) Unless term "average velocity" is used, we will assume velocity refers to instantaneous velocity. It is the slope of a tangent line to the position function... aka the derivative. On the velocity graph, we can also determine the direction an object moves: o When v(t) > 0, an object moves in a positive direction (up or right). o When v(t) < 0, an object moves in a negative direction (down or left). o When v(t) = 0, an object is stopped. • Velocity is a vector quantity and must have magnitude and direction. • Speed is the absolute value of velocity and is always positive
When f'(a) fails to exist graphically: corner
where the one-sided derivatives differ
When f'(a) fails to exist graphically: vertical tangent
where the slopes of the secant lines approach either ∞ or −∞ from both sides.
When f'(a) fails to exist graphically: cusp
where the slopes of the secant lines approach ∞ from one side and -∞ from the other
When f'(a) fails to exist graphically: discontinuity
which will cause one or both of the one-sided derivatives to be Non-existent.
Finding volume given square cross sections
x^2
Finding volume given isosceles right triangle with legs of length x as cross sections
x^2/2
Finding volume given rectangle cross sections with diagonals of length x
x^2/2
The derivative of y
y'
Finding volume given semicircle cross sections of radius x
πx^2/8
The limit of a function is:
• An output, y-value. output • A single, real value. • A y-value that may or may not be reached and we don't care! • An output the function gets really close to as the input gets really close to a number.
Fundamental Theorem of Calculus
∫ f(x) dx on interval a to b = F(b) - F(a)
Area of a Region Between Two Curves
∫(tom-bottom)dx or ∫(right-left)dy (the integrals from a to b)
Order of Integration
∫²₁ f(x) dx = -∫₁² f(x) dx
