APSC 171 – Calculus 1

APSC 171 – Calculus 1Elizabeth Lutton ([email protected])

OverviewDerivatives• Basic facts• Power, product, quotient, and chain rules• Implicit differentiation• Logarithmic differentiation• Linear approximationsIntegrals• Approximations (Reimann sums)• Solving using Fundamental Theorem of Calculus• Substitution• Improper integrals

Overview• Integration by parts• Partial fractions• Area between curves• Volume (slices and cylinders)• Work problems• Slope fields• Differential equationsLimits: L’Hôpital’s RuleParametric EquationsLast minute notes

Derivatives

DerivativesDerivatives to memorize:

Memory tip: notice that derivatives of “co” functions are negative and “non-co” functions

are positive

Derivatives

More derivatives to memorize:

Again notice the derivatives of “co” functions are negative and “non-co” functions are positive

Derivatives

Special cases because

And more…

DerivativesPower Rule:

Example:

DerivativesProduct Rule:

Example: = ?= =

“derivative of the first times the second,plus the first times the derivative of the second”

DerivativesExample: Simplify first!Write everything in terms of sine and cosine= = so we don’t even need the product rule=

DerivativesQuotient Rule:

Compare to Product Rule:

Product rule has + instead of – and no denominator

“derivative of the first times the second,MINUS the first times the derivative of the second,

all over the denominator squared”

DerivativesQuotient Rule:

Example: = =

DerivativesTip: The Product Rule is easier to use than the Quotient Rule, so you can raise the denominator to an exponent of -1 and then use the Product Rule instead of the Quotient Rule

Example:

same as answer from previous slide

DerivativesExample:

Simplify first! Factor the numerator = = = 2

DerivativesMoral of the story:

Before you do anything, check if you canSIMPLIFY, SIMPLIFY, SIMPLIFY!

Picture courtesy of http://www.zedge.net/profile/OoHumairahoO/

Derivatives

Chain Rule (used for compositions of functions):

f(x) is the “outer” function and g(x) is the “inner” function

Derivatives

Example: = = =

Tip: Rewriting square, cubic, etc. roots as numerical exponents when taking derivatives helps you to make fewer mistakes

Derivatives (Implicit)Implicit differentiation: what is it and when do you use it?• It’s used when y is a function of x instead of just

another variable: x and y are related in some way– Legs and dogs vs. cats and dogs

• Implicit differentiation is just a special case of the chain rule– 2y is a composition of functions where 2 is the

“outer” function and y is the “inner” function

Derivatives (Implicit)Example: Do you use implicit differentiation or not? How do you know?a) Find the tangent slopes of Yes, because y is obviously a function of x

b) No, because we are not told that y depends on x in any way; y seems to be a variable not a function. Therefore the answer to this problem is zero.

Derivatives (Implicit)

If in doubt, use implicit differentiation rather than not

Derivatives (Implicit)Example:Find the tangent slopes of (part a on previous slide) “tangent slopes” means

derivatives!

Derivatives

A note on notation: is a VARIABLE is a FUNCTION

is the same as y’(x) means “derive the following with respect to x”

Derivatives (Implicit)Example (question 3 on 2008/2009 final):Calculate the tangent slope to the curve at the point (1,2).

Note that you can substitute in values for x and y before solving for . It’s easier this way.

Derivatives (Implicit)

Therefore the tangent slope to the curve at (1,2) is

Derivatives (Logarithmic)Logarithmic differentiation: when do you use it?• Use when the variable with which you are

differentiating with respect to (usually x) is in the base AND the exponent

• e.g.

Derivatives (Logarithmic)Example (question 2 on 2010/2011 final):Find the derivative when

Derivatives (Logarithmic)Be sure to write everything in terms of x (and not y) in your final answer for logarithmic differentiation questions• See last two steps of previous slide

Derivatives (Linear Approximation)

Linear approximations use the y value f(a) at a point x=a, and the tangent line at x=a to approximate the value of a function


Tangent line to f(x) at x=a

(slope = f’(a))

Slope of tangent line at x=a+Δx

Slope of tangent line

at x=a≈


Memorize this equation:


Example: Approximate the value of 3+e0.2

0.2 is close to 0 so use a=0

Actual value:


Example: A circle has a radius of 12±0.3 mm. Estimate the maximum possible error in the area of the circle

the maximum possible error inthe area of the circle is mm

absolute value

because error is ±0.3

max error is 0.3

OptimizationTo solve optimization problems:1. Translate the given word problem into a

mathematical function f(x)2. Find the derivative of the function and set it

equal to zero to find the x values of the critical points

• i.e. the critical points are the solutions to f’(x)=0

3. Test the critical points using either the first or second derivative test to see if they are local maximums or minimums

Integrals

IntegralsIntegrals are “signed” or “net” area, which means they can have negative values

+

IntegralsThe integral on the previous slide could have also been calculated as follows:

Area of blue triangle – area of red triangle

Integrals (Approximations)Left Riemann sum:• Underestimate for increasing functions• Overestimate for decreasing functions

http://www.vias.org/calculus/img/04_integration-10.gif

Integrals (Approximations)

http://commons.wikimedia.org/wiki/File:Riemann-Sum-right-hand.png

Right Riemann sum:• Underestimate for decreasing functions• Overestimate for increasing functions


http://www.ltcconline.net/greenl/courses/116/IntegrationApps/riemann.htm

Midpoint Riemann sum:• A better approximation than left and right

Riemann sums

Integrals (Approximations)Trapezoidal sum:• Uses trapezoids rather than rectangles to

approximate the area under a curve• A better approximation than left, right, and midpoint approximations

http://www2.seminolestate.edu/lvosbury/CalculusI_Folder/Calc%20I%20exam4.htm

Simpson’s Rule:• Uses pieces of parabola to approximate the

area under a curve


http://eg1002.pbworks.com/w/page/6607488/Integration

Memorize this formula

IntegralsProperties of Integrals:

This is a rectangle where the width is zero

so the area must be zero

IntegralsMore properties of integrals:

where c is a constant

where

IntegralsWhat’s the difference between a definite and an indefinite integral?• A definite integral has bounds whereas an

indefinite integral does not• e.g. is a definite integral in an indefinite integral

Integrals• When solving indefinite integrals, find

the antiderivative of the function– where F(x) is the antiderivative of f(x) (i.e.

F’(x) = f(x))– Don’t forget the +C!!!!!!!!!!!!

• e.g. where C is a constant specific to the function

Integrals

Different functions with the same derivative (which is why we need

the +C term)

Integrals• When solving definite integrals, use the

second part of the Fundamental Theorem of Calculus This theorem is also helpful for physics

problems• e.g. where s(t) is the position function and

v(t) is the velocity function• Recall and

know this!

IntegralsExample (question 2 on 2010/2011 final): = Tip: sometimes it may be easier to see an antiderivative if you use negative exponents and rewrite square roots as fractional exponents= =

Integrals= = =

Integrals (Substitution)Method of Substitution• Use when you see a function and its derivative

in an integral– e.g. use substitution on because

• If solving an indefinite integral, don’t forget to convert back to the original variable– i.e. change u back to x

• If solving a definite integral, don’t forget to change the bounds of integration

Integrals (Substitution)Example: = = = =

Substitution:

+C!!!! Include constants in “u” because

so the constant “disappears” and the calculation is simpler

Integrals (Substitution)Tip: When checking your work for an integration problem, it’s faster and more effective to take the derivative of your answer and see if it matches the given problem, than to redo the integration

Integrals (Substitution)Example (question 2 on 2009/2010 final): = == = )=

Substitution:

Change bounds:

Integrals (Improper)Improper integrals have at least one of the following characteristics:1. One or both of the bounds is/are 2. The function has a discontinuity on the

interval you’re integrating over

Integrals (Improper)• An integral converges if the value of the

integral is finite• An integral diverges if the value of the

integral is infinite

Integrals (Improper)For integrals with bounds of ±∞, set up the integral as follows:

Integrals (Improper)

= = = = 0 + 1 = 1

Interval converges because 1 is finite

Means that the area under from x=0 to x=∞ is finite

Example:

Integrals (Improper)= = = = ∞

Interval diverges because the value of the integral is ∞

Means that the area under from x=0 to x=∞ is infinite

Example:

Integrals (Improper)For integrals with a discontinuity at x=z (where ), set up the integral as follows:

Integrals (Improper)If you split an integral into two integrals and , and either or (or both) diverges, then the original integral also diverges

Integrals (Improper) =

= = undefined

(because ln(0) does not exist… the domain of y=ln(x) is x>0)

Example:

Integrals (Improper)• When p>1, (the integral converges)• When p≤1, diverges

Memorize this fact!

Confirm using the example on the previous slide

Integrals (By Parts)Integration by Parts:

• Choose one part of given integral to be u and one part to be dv

• Be sure to include the dx (or dt) part in the dv term

∫𝜇∗𝑑𝑣=𝜇∗𝑣−∫𝑣 ∗𝑑𝜇memorize this!

Integrals (By Parts)Integration by Parts:• Usually dv is a function of sine and cosine

– Easy to integrate• Usually u has a power of x (x, x2, x9, ) or a

function of • These are very loose trends… don’t assume

they are always true• You will still get the same answer no matter

what you choose to be u and dv

Integrals (By Parts)Example (question 5a from 2008/2009 final): = = = = = = =

Substitution:Let

Let Let Integration by parts:

Integrals (Partial Fractions)Partial Fractions• Use when asked to find the integral of a

fraction that does not have a common factor in the numerator and denominator

• e.g.

don’t need partial fractions here because we had a common factor that simplified the problem

need partial fractions here because there is no common factor to simplify the expression

Integrals (Partial Fractions)• For partial fractions, you need to start with a

fraction in which the degree of the numerator is LESS than the degree of the denominator

• Use long division or synthetic division to turn an improper fraction into a polynomial plus a proper rational function

If you forget how to do synthetic division, it’s worthwhile to re-learn it. Check out this website:

http://www.purplemath.com/modules/synthdiv.htm



Integrals (Partial Fractions)Example (question 4a from 2008/2009 final exam): =

Forget about the integral for a minute = =

Integrals (Partial Fractions) = =

Integrals (Partial Fractions)

Integrals (Summary)Method of Solving an Integral

When to Use the Method

Reimann sums When you want an approximationSecond part of the Fundamental Theorem of Calculus

When you have a regular, “simpler” integral you can easily take the antiderivative of

Substitution When you see a function and its derivative in the same integral

Letting t approach ±∞ then taking the limit of the integral

When one or both of the limits is/are ±∞

Integration by parts When you have a product or quotient of unrelated functions (e.g. cos(x) and x3)

Partial fractions When you have an integral of a fraction that does not have a common factor in the numerator and denominator that can cancel to simplify the fraction

Integrals (Area)Area between curves:

∫𝑎

𝑏

[ 𝑓 (𝑥 )−𝑔(𝑥) ]𝑑𝑥

f(x) = the “top” functiong(x) = the “bottom” functiona = the leftmost intersection point of f(x) and g(x)b = the rightmost intersection point of f(x) and g(x)

Integrals (Area)

http://www.astarmathsandphysics.com/igcse_maths_notes/igcse_maths_notes_integration_and_the_area_between_two_curves.html

Blue curve is the top function and red curve is the bottom function

Green area= =

Integrals (Area)When integrating with respect to x (meaning you’re using “dx”), you are adding up infinite vertical strips of area

Integrals (Area)• Sometimes the top and bottom functions will switch• In this case you need to use two or more

integrals to find the area• Remember that integrals represent “net” or “signed

area” (i.e. they can be negative) and that area is always positive

𝝅𝟐

y=sin(2x)

y=cos(x)

𝝅𝟔

http://www.physicsforums.com/showthread.php?t=154524

Integrals (Area)Total area= + = +

y=sin(2x)

y=cos(x)

𝝅𝟐

𝝅𝟔

http://www.physicsforums.com/showthread.php?t=154524

Integrals (Area)We’re still adding up infinitely thin vertical strips of area

Integrals (Area)• Sometimes you have a “left” and a “right”

function rather than a “top” and a “bottom” function

• In this case, integrate with respect to y instead of x– i.e. use dy instead of dx

http://illuminasoftware.com/lesson10/Area_between_curves_print.html

“right” function(pink)

“left” function (blue)

Integrals (Area)Area between curves (with respect to y):

∫𝑎

𝑏

[ 𝑓 (𝑦 )−𝑔(𝑦 )]𝑑𝑦

f(x) = the “right” functiong(x) = the “left” functiona = the leftmost intersection point of f(y) and g(y)b = the rightmost intersection point of f(y) and g(y)

Integrals (Area)By integrating with respect to y, we’re now adding up vertical strips of area instead of horizontal strips

– Pancakes not books!

Integrals (Area)

http://www.horizonbook.com/http://stmaryslindsay.ca/events-in-the-parish/

pancake-and-sausage-brunch-fundraiser-sunday-jan-17-for-world-youth-day/attachment/pancakes-

2/

Vertical strips(dx)

Horizontal strips(dy)

Integrals (Area)Horizontal strips with thickness dy

Integrals (Area)Type of

functions Top and bottom Left and right

Graph

Integrate with respect to: x (use dx) y (use dy)

Horizontal or vertical strips vertical horizontal

Integrals (Volume)• When finding area, we used the fact

that (or dy)

• When finding volume, we will use the fact that (or dy)

• Width = dx or dy

Integrals (Volume)• There are two methods of calculating

volume: the slices method and the cylindrical shell method

• For slices, the “area” is the area of the circular slice () and the “width” is the thickness of the slice

• For cylindrical shells, the “area” is the surface area of the shell () and the “width” is the thickness of the shell

Integrals (Volume)Example (question 5a from 2008/2009 final):Find the volume of the solid generated by rotating the region bounded by x=(y-3)2 and the line x=4 about the line x=-1

1. Start by sketching the

region you want to revolve

We’ll use the slices method first

Integrals (Volume)

2. Reflect the area across the revolution line (in this case x=-1)3. Draw a sample slice of area (pictured in green)4. Label the points needed to find the radii (labeled in blue)

Integrals (Volume)5. Calculate the outer (r1) and inner (r2) radii

Integrals (Volume)6. Substitute radii terms into the volume equation

7. Evaluate the integral

Integrals (Volume)Tips for volume by slices problems:• Be sure to revolve the area around the

correct axis– This is important because it will affect your radii

functions

• Keep everything in the integral (i.e. bounds, radii functions, and dx or dy) in terms of only x or only y– Use or

Integrals (Volume)

http://urbanshakedowns.wordpress.com/2011/05/03/big-bangs-russian-dolls-infinity/

http://www.mrsjanuary.com/canada-deals/amazonca-4-piece-measuring-cups-set-50/

Cylindrical shells are infinitely thin concentric shells that fit inside each other like measuring cups or Russian dolls

Integrals (Volume)Let’s solve the previous problem using cylindrical shells

1. Again, start by sketching the region you want to revolve

Integrals (Volume)

2. Reflect the area across the revolution line (in this case x=-1)3. Draw a sample cylindrical shell (pictured in red)4. Label the points needed to find the radii (labeled in blue)

Notice that I had to rewrite the

equation of the parabola from to

Integrals (Volume)5. Calculate the radius and the height of the cylindrical shell

Integrals (Volume)6. Substitute radius and height terms into the volume equation 7. Evaluate the integral

This is the same answer that we found with the slices

method

Integrals (Work)Relevant equations:

mass volume density (kg) (m3) (kg/m3)

Force mass acceleration due to gravity (N) (kg) (m/s2)

work force displacement (J) (N) (m)

Integrals (Work)The equations on the previous slide lead us to this equation for work against gravity:

(see previous slide for units)

This is the equation to memorize!

work volume density gravity displacement

Integrals (Work)• Like area and volume problems, when solving

work problems we will need to add up an infinite number of infinitely small pieces of work to get the total work

• Also like before, this is done by using an integral

Integrals (Work)Example (question 13 from 2008/2009 final):A large cylindrical tank has radius 2 meters and height 5 meters. The tank has toppled over so that the valve, originally located in the centre of the base of the tank, is now 2 meters off the ground. The tank is half full of fuel oil. To put it back to its upright position, the oil has to be pumped out through the valve. Assuming that the density of the oil is 900 kg per cubic meter, and that the acceleration due to gravity is g=9.8m/s2, what is the (minimal) amount of work required to empty the tank?

Integrals (Work)In other words:How much work is required to pump oil out of a sideways cylinder with the following parameters: r=2m, h=5m, ρ=900 kg/m3, and g=9.8m/s2? The tank is half full and the oil needs to be pumped out of a valve that is in the centre of one side.

Integrals (Work)

tank is only half full so the volume of oil in tank = half of

the volume of the cylinder

displacement (infinitely small)

h = 2-y(see picture)

Integrals (Work)

π, ρ, r, and g are all constants so pull them

out in front of the integral

If units are given in the problem definition then remember to give

them in your answer!

Differential Equations• A differential equation is an equation

involving derivatives• “Solving” a differential equation means

finding the original function (usually x(t)) that satisfies the given conditions– e.g. a solution of is because

Differential EquationsSpring systems yield differential equations Newton’s 2nd law: Hooke’s Law: (k = spring constant)

Recall that acceleration is the 2nd derivative of

positionLet

Differential EquationsThe solution of can be written in the form or where A, B, C, and φ are arbitrary constants

Differential Equations (Slope Fields)

A slope field is “a graphical representation of the solutions of a first-order differential equation”1

Looks like because it is!

1 http://en.wikipedia.org/wiki/Slope_fieldhttp://www.math.rutgers.edu/~sontag/JODE/JOdeApplet.html


To draw slope fields, first evaluate at each (x,y) coordinate then draw a little line on that point with a slope equal to the value of you calculated.


Example: draw a slope field for Evaluate at each point Some sample calculations:


Example: draw a slope field for

http://www.math.rutgers.edu/~sontag/JODE/JOdeApplet.html

Differential Equations (Solving)

A separable differential equation is a differential equation that can be written in the form


To solve a separable differential equation:1. Move all the terms involving y to one side

of the equation and all of the terms involving x to the other side

2. Integrate each side of the equation separately– Don’t forget the +C !

3. Isolate for y


Let’s solve the differential equation from the previous example:

It doesn’t matter whether you put the constant of on the x or the y side of the

equation

This is the general solution. Solve for C2 if given an initial

condition


Slope field of initial differential equation

Graph of final solution (where

C2=0)

http://rechneronline.de/function-graphs/http://www.math.rutgers.edu/~sontag/JODE/JOdeApplet.html

Differential Equations (Logistic Function)

• The logistic equation is often used to model population growth (opposed to the exponential model, which is unrealistic)

http://www.squarecirclez.com/blog/h1n1-and-the-logistic-equation/3498

This is the graph of the function P(t) NOT the graph of the differential

equation


• The logistic differential equation is • P is population K is carrying capacity (constant) k is a constant t is time


Solving the logistic differential equation: Isolate for y

Use partial fractions: and

Limits (L’Hôpital’s Rule)L’Hôpital’s Rule:• Use for indeterminate limits (i.e. limits

that equal )– Note that are not the only indeterminate

forms that exist– ’s are not created equal so !

Limits (L’Hôpital’s Rule)• L’Hôpital’s Rule says if and then

– This is NOT the quotient rule!• To use L’Hôpital’s Rule, make sure you have an

indeterminate limit, then take the derivative of the numerator and the derivative of the denominator and try taking the limit again

• Sometimes you need to use L’Hôpital’s Rule twice for the same limit

Limits (L’Hôpital’s Rule)Example (question 3 from 2010/2011 final): = = = = 12

Use L’Hôpital’s Rule

Use L’Hôpital’s Rule again

Parametric Equations

is the same as…

…which is the same as…

Parametric EquationsTo easily graph a vector valued function, eliminate the parameter t using algebra• Isolate t in one of the equations x(t) or y(t),

then substitute it into the other equation

Parametric EquationsExample: find the equation that is not dependent on t that describes the path of a particle for which and .

Isolate t in the x(t) equation to get The y equation now becomes

Therefore the particle follows a parabolic path

Parametric Equations• The position vector has velocity vector and

acceleration vector • The speed of a particle is the length of its

velocity vector

Parametric EquationsWhat is the velocity and speed of a particle whose position is described by when t=1?

Speed = length of velocity vector =

If units are given in the problem statement, remember to give them in your answer!

Last Minute Notes

Random trig facts:

Last Minute NotesHelpful sources:• www.wolframalpha.com

– Calculations, graphs, etc.• www.khanacademy.org/

– Videos showing you how to solve various types of problems (e.g. implicit differentiation, integration by parts, etc.)

• Me! Email me at [email protected]

http://www.wolframalpha.com/

http://www.khanacademy.org/

APSC 171 – Calculus 1

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