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University of Houston

Mathematics

MATH 4389

Almus

Spring

Description

Calculus 1 Review
Limit and Continuity
Differentiation
Applications of Derivatives
Exponential and Logarithmic Functions
Integration
Must know all rules of differentiation
Rate of change, related rates, optimization.
Increasing/Decreasing Function, local/absolute extreme values, concavity
Gathering info from the graph of f, or f’ or f’’.
L’Hospital’s rule.
Understand how integrals are defined, know all basic rules of integration, u-sub.
Intermediate Value Theorem, Extreme Value Theorem, Rolle’s Theorem, Mean Value Theorem
Fundamental Theorem of Calculus Definition of derivative:
Equation of tangent line at (a,f(a)) : yf f(x)a(') )
Rules of differentiation: uv '' u' v uv
v uvuu ''
' 2
vv
) ( ' ) ) ( ( )' ( ' () f gx fg xgx
Newton’s Method
Goal: To approximate a solution to f()0 (a root or a zero of the function
f(x)).
Start with a guess;0
If f 0 , then next guess is: f 0x
0 1 0 f x0)
In general:xn n f )n
f xn) Gathering information from the derivative function:
Example: Given the graph of f '(x.
Exponential growth/decay
kt
P(t) Pe0
Half life: kT ln(2)
kT ln(2)
Doubling time: Riemann Sums
Recall – left endpoint, right end point, midpoint, Trapezoid approaches, Upper
sum, Lower sum. You need to know how to put them in order depending on
whether the function increases/decreases.
Example: Given the definite integral, how do these sums compare with it?
Know the definition of a definite integral.
Note: For a positive function, definite integral gives the area under the curve.
If the formula of a function is not given, but the graph is given, you can use the
area under that function to find the definite integral. 6
Example: The graph given below belongs to f x). fxd) ?
0
Basic rules of integration
TABLE OF INTEGRALS
r xr1 1
x dx C ; r 1 dx ln x C
r 1

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