Dierentiation The Easy Way
Math 1a
Fall, 2011
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Some Dierentiation Rules
Suppose c is a constant and f and g are dierentiable functions.
d
f (x) g(x) = f (x) g (x)
1 The derivative of a sum or dierence:
dx
2
The derivative of a constant multiple of a fu
RATES OF CHANGE IN NATURAL AND SOCIAL SCIENCES
Example 1. Suppose that C(s) gives the number of calories that an average adult burns by walking at a steady speed of s miles per hour for one hour. dC (1) What are the units of ? ds dC is Solution Because C
Math 1A: introduction to functions and calculus
Oliver Knill, 2014
Lecture 20: Worksheet
Anti derivatives
Here are some trickier anti derivative puzzles. We still have no integration techniques and must rely on intuition and experiments to nd
the derivati
Limits of Functions
Math 1a
1
Fall, 2012
The graph below shows the plot of a function y = f (x). Determine whether the limits shown
below exist. If the given limit does exist, nd this limit.
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12 .
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Math 1a
1
The Intermediate Value Theorem
Fall, 2012
Some discussion questions:
(a) Were you ever exactly three feet tall?
(b) Was your height (in inches) ever the same as your weight (in pounds)?
(c) Are there two points on exactly opposite sides of the w
Math 1a
1
The Denition of Derivatives
For each of the following functions f (x) and points a, nd the slope of the tangent line to
y = f (x) at x = a, then nd the equation of that tangent line.
(a) f (x) = 2x + 7 at x = 1
(b) f (x) = x2 at x = 2
(c) f (x)
Innite Limits
Math 1a
1
Try to estimate the limit lim
+
x3
x
4
Fall, 2012
x4
by creating a table of values.
x3
3.5
3.1
3.01
3.001
3.0001
x4
x3
2
Using the idea of the previous problem (but without creating a table), calculate the following
limits:
(a) lim
Implicit Dierentiation
Math 1a
Fall, 2012
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Often functions are dened explicitly, like y = x3 ex or y = cos(x + 1). But sometimes
functions are dened implicitly, like the circle x2 +y 2 = 1. This is really two functions
y = 1 x2
and
y = 1 x2 .
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Of course
Math 1a
Related Rates
Fall, 2012
1
Two cars are approaching an intersection. A red car, approaching from the north, is traveling
30 feet per second and is currently 60 feet from the intersection. A blue car, approaching from
the east, is traveling 20 feet
Calculating Denite Integrals
Math 1a
Fall, 2012
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The Denite Integral
If F (x) is any anti-derivative of f (x) (that is, if F (x) = f (x),
then the denite integral of f (x) from a to b is
b
b
= F (b) F (a).
f (x) dx = F (x)
a
a
&
%
Compute the following
Substitution
Math 1a
Fall, 2012
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The Substitution Rule:
Suppose u = g(x) is a dierentiable function with domain an interval I and f (x) is continuous
on I. Then
f (g(x)g (x) dx = f (u) du.
Thus we can make substitutions and treat du and dx like dierent
Math 1a
Optimization
Fall, 2012
1
Suppose a rectangular region has xed perimeter of 40 cm. What is the largest area the region
can have?
2
I have some old fencing and want to use it to fence in a rectangular region against my house
for my new pet bunnies.
Integration Applications
Math 1a
1
In this problem we will nd the area between the curves
y = f (x) = x2 4
and
y = g(x) = 2x 1,
shown at right.
(a) These two curves intersect at (x, y) = (1, 3) and
(3, 5). Cut the interval [1, 3] into n subintervals
and a
The Denite Integral
Math 1a
Fall, 2012
We dene the denite integral of y = f (x) from x = a to x = b as
n
b
f (x )x
i
f (x) dx = lim
n
a
i=1
where xi1 x xi . Note that x could be xi (in which case we have the limit of Rn ) or xi1 (in
i
i
which case we have
Math 1a
The Fundamental Theorem, Part One
Fall, 2012
x
Today well be concerned with the area function F (x) =
f (t) dt, given by the shaded area here
a
under the curve y = f (t):
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Maxima and Minima
Math 1a
Fall, 2012
For each of the following graphs, mark the points that are local maxima and local minima inside
the interval [a, b]. (A function f (x) has a local maximum at c if f (x) f (c) for all x in a small
open interval containi
Compositions of Functions
Math 1a
1
Each of the following is a composition of some of the three functions
f (x) = sin(x),
g(x) = x2
and
h(x) = x + 1.
Please identify the composition.
(a) sin(x2 )
(c) x2 + 1
(d) sin2 (x + 1)
(e) sin(x + 1)2 )
(f) (sin(x) +
Limits To Innity
Math 1a
1
Fall, 2012
Guess (eye-ball) each of limits listed below.
x2 1
x x2 + 1
(a) lim
(d) lim
x
x2 5 x
x1
x x2 + 1
(b) lim
(e)
lim
x
x + x2
(c)
x2 + 1
x x 1
lim
x3 + 27x 1
x
2x3 + x2
(f) lim
2
Find each limit in Problem 1 using algebra
More On Derivatives
Math 1a
1
Fall, 2012
For each of the following functions f (x), use the denition of derivative
f (x + h) f (x)
h0
h
f (x) = lim
to compute the derived function f (x).
(a) f (x) = 2x + 7
(c) f (x) = 2 x
2
(b) f (x) = x2
(d) f (x) =
1
x
Math 1A: introduction to functions and calculus
Oliver Knill, 2014
Lecture 17: Worksheet
Catastrophes
We see here graphs of the function f (x) = x4 cx2 for c between 0
and 1:
1
Draw the bifurcation diagram in this case. The vertical axes
is the c axes.
Da
Math 1A: introduction to functions and calculus
Oliver Knill, 2014
Lecture 16: Worksheet
Theorems, Theorems, Theorems
1
Formulate the mean value theorem.
2
Formulate the intermediate value theorem.
3
Formulate the Rolles theorem.
4
Formulate the extreme v
Math 1a
1
Velocities, Secants & Tangents
Fall, 2012
My father lives a two-and-a-half hour (150 minute) drive away. On a recent trip to visit him
I recorded the trip odometer at regular intervals:
Time (minutes) 0 30 60 90 120 150
Distance (km)
0 30 80 135