CALCULUS
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Calculus is the study of change.
Ex: displacement, velocity, etc
It is used in biology, chemistry, physics, economics, etc.
Instantaneous rate of change
→ means the rate of change for a split second in time (or one specific point on a
curve)
Recall: A tangent is a line that cuts a curve or a circle in exactly one place.
A secant is a line passing through any 2 points on a curve.
Given a curve f and a point on the curve P(x, f(x)) there exists another point Q on the
curve such that PQ is a secant to f.
Imagine we want to find the instantaneous rate of change at the point P.
In general, rate of change is equal to the gradient (slope),
so that we need find the gradient at P.
We cannot find the slope of a curve, so we use the secant to estimate.
If we make h a smaller value (ie: closer to P) we get a better approximation.
By letting h get smaller and smaller until h is so small that we have a tangent.
As the distance between P and Q is reduced, PQ approaches 0, forming a tangent line
at P.
Thus as h0, the tangent is formed.
y 2  y1
x 2  x1
The slope of the tangent or the instantaneous rate of change is:
The slope of PQ 
Lim
h0
f(x + h) – f(x)
h
This definition is called a derivative or Newton’s Quotient.
It is a comparison of how much y changes compared to how much x changes at any
instant on the curve.
A derivative represents the
slope of the tangent at a given
point.
Ex: Find the instantaneous rate of change for the curve f(x) = x2 at the point x = 2
Slope =
Lim
h0
f(x + h) – f(x)
h
Ex: Calculate the derivative of f(x) = 3x2 – 6x + 4 at x = 1.
Ex: Find the equation of the tangent to f(x) = x3 – 1 at the point x = -2.
Ex: Find the derivative of y = x3 + x.
DIFFERENTIATION
Differentiation is the process of finding a derivative
The derivative of a function at a point, f(x), is the gradient of the tangent line at
that point.
Notation:
1. f `(x)
→ “f prime of x”
2. y` → “y prime”
or
“f dash of x”
3. dy
→ Leibniz notation
dx →“the derivative of y in terms of x”
→ read as “d y d x”
4. d → read as “d d x”
dx
Differentiation rules
We can use Newton’s Quotient to find a derivative, but it takes too long.
There are a few rules to help!
1.
The Constant Rule
If f(x) = c (where c is any constant),
then f `(x) = 0
Ex: f(x) = 2
dy = 0
dx
A constant is a horizontal line. Any tangent would lie directly on top of it.
The slope of a horizontal line is always 0.
2. The Power Rule
If f(x) = xn, then f `(x) = nxn-1.
(In plain English, bring the exponent down in front and reduce the new
exponent by 1.)
Ex:
1. f(x) = x2
dy =
dx
f `(x) =
2. f(x) x100
3. f(x) =
1
x3
d=
dx
3. The Constant Multiple Rule
If f(x) = kg(x), then f `(x) = k(g`(x)).
Ex: f(x) = 8x3
f `(x) =
Ex: g(x) = -3x15
g`(x) =
4. The Sum and Difference Rule
If f(x) = g(x) + h(x), then f `(x) = g`(x) + h`(x)
If f(x) = g(x) - h(x), then f `(x) = g`(x) - h`(x)
Ex:
a)
f(x) = 2x2 + 3x
b)
c)
f(x) = 7x3 – 5x2 + 3x – 2 f `(x) =
y = 4x - x2 + 1
f `(x) =
y`=
Application:
Determine the coordinate(s) on the curve x:→x3 – x + 2 where the gradient is 11.
BE SMART!
Questions like y = (3x2)4 and f(x) = (x – 3)2 is must be put in power form before you
take the derivative If it is easier to multiply out the expression – save yourself some
precious seconds!!
NATURAL LOG DERIVATIVES
Dx ex = ex