Introduction to Calculus and Study Guides

I have to admit; I was one of those in high school and even college who never really “got” calculus.  I could go through the motions of doing really hard problems, but most of the time, never really understood why I was doing them.

Calculus can be that way; and sometimes it’s all right!  I hope to be able to help you know what you need to understand and what you just need to be able to do, when going through your Calculus classes.  Honestly, there are some parts that I still don’t really understand.

This first section of Calculus will cover the Advanced Placement (AP) AB course, since that’s what I typically tutor.  I hope to eventually cover the more advanced BC course, and I will indicate it as so.

To simplify, typically your first calculus class has to do with rates of things (Differentiation) and area of things (Integration).  What calculus adds to what you’ve been doing in Algebra and Geometry is that the concepts are extended to finding rates and areas of curvy things (lines and 3-D objects); it was first invented in the 1700’s (possibly earlier!) to be used in many concepts in Physics (not my favorite subject!) J

In fact, calculus grew from some problems that European mathematicians were working on during the seventeenth century:  general slope, or tangent line problems, velocity and acceleration problems, minimum and maximum problems, and area problems.

Calculus usually starts out with the study of limits, which can be quite frightening and confusing.  This is because we need limits to “understand” the foundation of Differential Calculus; we need to “appreciate” the math before we plug away at doing it much easier.  Let’s plug through limits as well as we can!

Here is a Study Guide for Differentiation:

CALCULUS CHEAT-SHEET: DIFFERENTIATION

DERIVATIVES:

 

Definition of Derivative:

\(\displaystyle {f}’\left( x \right)=\underset{{h\to 0}}{\mathop{{\lim }}}\,\frac{{f\left( {x+h} \right)-f\left( x \right)}}{h}\)

 

Derivative at a Point:

\(\displaystyle {f}’\left( c \right)=\underset{{x\to c}}{\mathop{{\lim }}}\,\frac{{f\left( x \right)-f\left( c \right)}}{{x-c}}\)

 

Constant Rule:

\(\displaystyle \frac{d}{{dx}}\left( c \right)=0\)

 

Power Rule:

\(\displaystyle \frac{d}{{dx}}\left( {{{x}^{n}}} \right)=n{{x}^{{n-1}}}\)

 

Product Rule:

\(\displaystyle \frac{d}{{dx}}\left( {f\cdot g} \right)=f\cdot {g}’+g\cdot {f}’\)

 

Quotient Rule:

\(\displaystyle \frac{d}{{dx}}\left( {\frac{f}{g}} \right)=\frac{{g\cdot {f}’-f\cdot {g}’}}{{{{g}^{2}}}}\)

 

Chain Rule:

\(\displaystyle \frac{d}{{dx}}\left( {f\left( u \right)} \right)={f}’\left( u \right)\cdot {u}’\)

 

Trig Derivatives:

\(\displaystyle \begin{array}{l}\frac{d}{{dx}}\left( {\sin \left( x \right)} \right)=\cos \left( x \right)\\\frac{d}{{dx}}\left( {\cos \left( x \right)} \right)=-\sin \left( x \right)\\\frac{d}{{dx}}\left( {\tan \left( x \right)} \right)={{\sec }^{2}}\left( x \right)\\\frac{d}{{dx}}\left( {\cot \left( x \right)} \right)=-{{\csc }^{2}}\left( x \right)\\\frac{d}{{dx}}\left( {\sec \left( x \right)} \right)=\sec \left( x \right)\tan \left( x \right)\\\frac{d}{{dx}}\left( {\csc \left( x \right)} \right)=-\csc \left( x \right)\cot \left( x \right)\end{array}\)

 

(Remember: Trig functions starting with “c” are negative. I also remember that there are always 2 sec’s (csc’s) and 1 tan (cot’s) in trig derivatives.)

Inverse Trig Derivatives:

\(\displaystyle \begin{align}\frac{{d\left( {\arcsin u} \right)}}{{dx}}&=\frac{{{u}’}}{{\sqrt{{1-{{u}^{2}}}}}}\\\frac{{d\left( {\arctan u} \right)}}{{dx}}&=\frac{{{u}’}}{{1+{{u}^{2}}}}\\\frac{{d\left( {\text{arcsec}\,u} \right)}}{{dx}}&=\frac{{{u}’}}{{\left| u \right|\sqrt{{{{u}^{2}}-1}}}}\\\frac{{d\left( {\arccos u} \right)}}{{dx}}&=\frac{{-{u}’}}{{\sqrt{{1-{{u}^{2}}}}}}\\\,\frac{{d\left( {\text{arccsc}\,u} \right)}}{{dx}}&=\frac{{-{u}’}}{{\left| u \right|\sqrt{{{{u}^{2}}-1}}}}\\\frac{{d\left( {\text{arccot}\,u} \right)}}{{dx}}&=\frac{{-{u}’}}{{1+{{u}^{2}}}}\end{align}\)

 

Exponential and Log Derivatives:

(u is function of x, a is constant)

\(\displaystyle \begin{align}\frac{d}{{dx}}\left( {\ln u} \right)&=\frac{{{u}’}}{u}\\\frac{d}{{dx}}\left( {{{{\log }}_{a}}u} \right)&=\frac{{{u}’}}{{u\left( {\ln \,a} \right)}}\\\frac{d}{{dx}}\left( {{{e}^{u}}} \right)&={{e}^{u}}{u}’\\\frac{d}{{dx}}\left( {{{a}^{u}}} \right)&=\left( {\ln \,a} \right){{a}^{u}}{u}’\end{align}\)

\(\displaystyle \frac{d}{{dx}}\left[ {f{{{\left( x \right)}}^{{g\left( x \right)}}}} \right]:\text{take ln of each side}\)

When we have a variable both in the base and the exponent, take ln of both sides to take derivative, and use implicit integration. Then substitute y function back in to get in terms of x.

 

Mean Value Theorem:

If a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there is at least one number c in (a, b), where

 \(\displaystyle {f}’\left( c \right)=\frac{{f\left( a \right)-f\left( b \right)}}{{b-a}}\).

(You may have to find the “c” by taking the derivative, setting to the slope you get on \(\left[ {a,\,b} \right]\) and solving for “c” in \(\left[ {a,\,b} \right]\).)

Intermediate Value Theorem (IVT):

If a function f is continuous on a closed interval \(\left[ {a,\,b} \right]\), where \(f\left( a \right)\ne f\left( b \right)\), and m is any number between \(f\left( a \right)\) and \(f\left( b \right)\), there must be at least one number c in \(\left[ {a,\,b} \right]\) such that \(f\left( c \right)=m\).

 

Rolle’s Theorem:

If a function is continuous on a closed interval and differentiable on the open interval (a, b), and  \(f\left( a \right)=f\left( b \right)\) (the y’s on the endpoints are the same), then there is at least one number c in (a, b), where \({f}’\left( c \right)=0\).

 

Curve Sketching:

Critical Point(s): point(s) where derivative is zero or undefined (critical points could also be endpoints)

Local Minimum: \(\frac{{dy}}{{dx}}\) goes from negative to 0 (or undefined) to positive, or \(\frac{{{{d}^{2}}y}}{{d{{x}^{2}}}}>0\)

Local Maximum: \(\frac{{dy}}{{dx}}\) goes from positive to 0 (or undefined) to negative, or \(\frac{{{{d}^{2}}y}}{{d{{x}^{2}}}}<0\)

Point of Inflection (concavity changes): \(\frac{{{{d}^{2}}y}}{{d{{x}^{2}}}}\) goes from positive to 0 (or undefined) to negative, or negative to 0 (or undefined) to positive.

When looking at graphs:

Function

First

Derivative

Second

Derivative

P

Point of

Inflection

(POI)

M

Maximum

or

Minimum

S

Sign Change

Max: + to ­­­­­–

Min: ­ – to +

Instantaneous vs. Average Rate of Change:

Instantaneous rate of change (e.g., a velocity) between two points is the slope of the tangent line, which is the derivative at a point. Average rate of change over \(\left[ {a,\,b} \right]\) is the slope of the secant line, which is \(\frac{{f\left( b \right)-f\left( a \right)}}{{b-a}}\).

 

Equation of Tangent Line at a Point:

  1. Take the derivative of the function; this is the slope of the tangent line. Plug the “x” value of the point given into the derivative to get the actual (numerical) slope “m“.
  2. Write the equation of the tangent line in the form \(y-{{y}_{1}}=m\left( {x-{{x}_{1}}} \right)\), using the original point \(\left( {{{x}_{1}},\,{{y}_{1}}} \right)\). (You may need to get the “\({{y}_{1}}\)” value using the original function, if only “\({{x}_{1}}\)” value is given). Simplify, if needed.

Point of Horizontal (Vertical) Tangent Line:

  1. Take the derivative, and set the numerator to 0 (horizontal tangent) or denominator to 0 (vertical tangent).
  2. Solve for x, and then get y from plugging in x into the original function.

Local (Tangent Line) Linearization:

  1. Find a “close” point that works with the original function. For example, for \(f\left( x \right)=\sqrt{x}\), to find \(\sqrt{{4.02}}\), use point \(\left( {4,\,2} \right)\).
  2. Take the derivative of the function; this is the slope of the tangent line. Plug the “x” value of the point from above (“4”) into the derivative to get the actual (numerical) slope “m“.
  3. Write the equation of the tangent line in the form \(y-{{y}_{1}}=m\left( {x-{{x}_{1}}} \right)\), using the original point found above, and the slope found in step 2 (for example, \(y-2=m\left( {x-4} \right)\). Simplify, if needed. Then plug the actual x (for example, 4.02) to get the actual y.

Optimization:

  1. Solve in terms of one variable, and take the derivative; set to 0 to get the minimum or maximum.  
  2. In the case of a closed interval, check the endpoints of the interval to make sure they aren’t lower or higher than the minimum or maximum found.

Implicit Differentiation:

  1. Differentiate both sides of equation with respect to x. When are you differentiating variables other than x (such as “y”), remember to multiply that term by \(\frac{{dy}}{{dx}}\text{ (}{y}’)\).
  2. Move \({y}’\) to the left side of the equation and move all other terms to the right side (even if they have x’s and y’s in them).
  3. Factor out the \({y}’\) on the left side of the equation to isolate and solve for \({y}’\).

Related Rates Hints:

  1. Draw a picture, and label any amounts that could be changing (variables), and not changing (constants).
  2. Write down exactly what the problem gives you, and what you need to solve for. For example, you may write down “Find \(\frac{{dA}}{{dt}}\) when radius = 6”. The rates (things that are changing) have “dt” (with respect to time). For rates, make sure the sign is correct (for example, “filling up” is positive a volume rate, “emptying out” is a negative volume rate).
  3. Write an equation that relates all of the given information and variables.
  4. Simplify by trying to put everything in as few variables as possible before differentiating (you may have to substitute some variables by solving in terms of other variables). Typically, when you have values that say “when” something happens, these are put in after differentiating.
  5. Use implicit differentiation to differentiate with respect to time.  Fill in the equation with what you know (for example, the “when”s).
  6. Solve the equation for what the problem is asking for.

Error Approximation:

To estimate the error (or % error) of a measurement,

  1. Attack this like a related rates problem: write down what we know, what we need, and how we relate the variables, such as with a Geometry formula, like volume.
  2. Use implicit differentiation to take the derivative of the function given; the error given will be one of the dx parts of the derivative (for example, change in volume might be dV).
  3. Solve for the error of the measurement needed, such as ds. To get % error, divide by the total amount of this measurement.

Position, Velocity, and Acceleration:

  • If a particle is moving along a horizontal line, its position (for example, relative to the origin) is a function, the derivative of this function is its (instantaneous) velocity, and the derivative of its velocity is its acceleration (how fast its velocity is changing). Displacement is how far the particle is from its original position.
  • Position, velocity, acceleration and displacement are vectors. Distance is the absolute value of the displacement vector and speed is the absolute value of the velocity vector; these are scalars.
  • If the derivative (velocity) is positive, the object is moving to the right (or up, if that’s how the coordinate system is defined); if negative, it’s moving to the left (or down); if the velocity is 0, the object is at rest.  This is also called the direction of the object.
  • Average Velocity is \(\frac{{\text{Ending Position }-\text{ Initial Position}}}{{\text{Total}\,\,\text{Time}}}=\frac{{\Delta x}}{{\Delta t}}\).

Derivative of an Inverse Function:

Let \(f\left( x \right)\) be a function that is differentiable on a certain interval. If \(f\left( x \right)\) has an inverse function \(g\left( x \right)\), and \(g\left( x \right)\) is differentiable for any value of x such that \({f}’\left( {g\left( x \right)} \right)\ne 0\), then \({g}’\left( x \right)=\frac{1}{{{f}’\left( {g\left( x \right)} \right)}}\).

(If we want to find the derivative of the inverse of the function at a certain point “x”, we just find the “y” for the particular “x” in the original function, and use this value as the “x” in the derivative of this function. Then take the reciprocal of this number; this gives to get the derivative of the inverse of the original function at this point).  

 

 

On to Limits and Continuity!

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