The Product Rule and the Quotient Rule
Product Rule
If f(x) and g(x) are both differentiable, then (f(x)g(x) = f(x)g(x)+f(x)g(x)
(f(x)g(x) = F(x) = limh->0(F(x+h)-F(x)/(h) = limh->0(f(x+h)g(x+h)-f(x)g(x)/(h)
Add and subtract f(x+h)g(x) to the numerator
F(
Fall 2015
Math 1060/1061, Calculus I
Exam III review
The problems in this review are representative of the types of questions that may be asked on the test. However,
they do not exhaust all possible exam questions; in turn, the exam may not include some t
3.9 Related Rates
Idea: Solve for a rate of change of one quantity in terms of another rate of change.
Example:
How quickly is the volume of a cube increasing if the side lengths are increasing at a rate of 2
cm/s and its current side length is 4 cm.
Let
First Day of Calculus:
Section 2.1 The Tangent and Velocity Problems
Definition:
A secant line to a curve is a line that passes through two points on that curve. Graphically, this
looks like:
A tangent line to a curve is a line that touches the curve, but
2.5 Continuity
Definition: A function f(x) is continuous at a number a if all 3 conditions are satisfied:
1. f(a) is defined
2. limx->af(x) exists
3. limx->a f(x) = f(a)
Note: If condition 3 holds, then 1 and 2 must hold.
Example:
Prove that f(x) = x2+3 i
Applications of the Chain Rule
Alternate Proof for the Quotient Rule
Use the Chain Rule and the Product Rule to prove the Quotient Rule
(f(x)/(g(x) = (f(x)*g(x)-1) = f(x)*g(x)-1+f(x)*(g(x)-1)
= f(x)*g(x)-1 + f(x)*(-g(x)-2)*g(x)
= (f(x)/g(x) - (f(x)*g(x)/(
Power Law:
limx->a(f(x)n = (limx->af(x)n if n is a positive integer
Root Law:
limx->a f(x) to the nth root is equal to (limx->af(x) to the nth root if n is a positive integer.
Direction substitution Property
If f(x) is a polynomial or a rational function
Special Trigonometric Limits
The function sin(x) satisfies limx->0(sin(x)/(x) = 1
For proof, see page 191 in the textbook or check in Course Documents on Blackboard.
The cos(x) function satisfies limx->0(cos(x)-1)/(x) = 0
Proof:
limx->0(cos(x)-1)/(x) = li
3.4 The Chain Rule
If g(x) is differentiable at x and f(u) is differentiable at u = g(x), then the derivative of F(x) =
f(g(x) is F(x) = f(u)*g(x) = f(g(x) * g(x)
In Leibniz notation, if y = F(x), then dy/dx = dy/du * du/dx
Proof is on page 203 in text.
E
3.6 Derivatives of Logarithmic Functions
Recall the following properties of logarithms:
ln(xy) = ln(x)+ln(y)
ln(x/y) = ln(x)-ln(y)
ln(xn) = nln(x)
for x>0, y>0, and n for all real numbers
Also, recall the derivative d/dx(ln(x) = 1/x for x>0
Consider ln(-x
Exponential and Logarithmic Functions
Definition: An exponential function is a function of the form f(x) = bx where b > 0.
Laws of Exponents
If a>0, b>0, and x and y are real numbers, then bx+y = bxby, bx-y = (bx)/(by)
(bx)y = bxy, (ab)x = axbx
Definition
3.5 Implicit Differentiation
Example
Find the tangent lines that meet a circle with radius 2 that is centered at the origin, at x=1.
Equation of the circle is x2+y2=4
Method 1 - Solve for y
y = cfw_sqrt(4-x2) for upper semi-circle
cfw_-sqrt(4-x2) for lowe
2.6 Limits at Infinity
Notation:
Let f(x) be a function defined on some interval (a, infinity). If f(x) approaches a number L as x
increases, then we write limx->infinityf(x) = L. If f(x) approaches as x approaches negative infinity,
we write limx->-infin
2.7 Derivatives and Rates of Change
Recall from section 2.1, (Lecture 1) that we used the slope of a secant line over a small interval
to approximate the slope of a tangent line.
Now we have the tools to compute the slope of a tangent line exactly.
Defini