math 123 midterm 1 tutorial surath gomis saskatoon engineering students society october 7, 2015
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OVERVIEW Preliminaries Limits Continuity Tangents DifferentiationTRANSCRIPT
MATH 123 MIDTERM 1 TUTORIAL
Surath GomisSaskatoon Engineering Students’ Society [email protected] 7, 2015
OVERVIEW Preliminaries Limits Continuity Tangents Differentiation
PRELIMINARIESComposite function:The functional argument becomes another function. The domain may look different upon inspection.
EXAMPLE 1 If , calculate and specify its domain.
EXAMPLE 2
cos ( 3𝜋4 + 𝜋6 )
LIMITS Limits define the behaviour at boundaries. This behaviour can be different depending on what side you approach from.
EXAMPLE 3 Evaluate If it does not exist, is the limit infinity, negative infinity, or neither?
EXAMPLE 4
lim𝑦→ 1
𝑦−4 √𝑦+3𝑦2−1
EXAMPLE 5
lim𝑥→0
|3𝑥−1|−∨3 𝑥+1∨ ¿𝑥 ¿
EXAMPLE 6
lim𝑥→∞
3 𝑥3−5𝑥2+78+2 𝑥−5 𝑥3
EXAMPLE 7
lim𝑥→∞
𝑥2+sin (𝑥 )𝑥2+cos (𝑥)
EXAMPLE 8
lim𝑥→∞
𝑥√𝑥+1(1−√2𝑥+3)7−6 𝑥+4 𝑥2
EXAMPLE 9
lim𝑥→ 3
13−𝑥
CONTINUITY Yet more ways to observe the behaviour of functions. Continuity is just as it sounds – a function which traverses without breaks or tears in smoothness is continuous.
EXAMPLE 10 Make the following continuous at the given point: at
EXAMPLE 11 Show that the function has the value (a+b)/2 at some point x. (IVT problem).
TANGENTS More behaviour... One can study a function by knowing its slope. This is crucial for engineering, for rates of change (i.e. slope) help us determine how systems operate in time.
Definition of the derivative:
EXAMPLE 12 Find the equation of a straight line tangent to at (2,3), using the definition of the derivative.
EXAMPLE 13 Find all points on the curve where the tangent line is parallel to the x-axis.
EXAMPLE 14 For what value of constant k do the curves and intersect at right angles?
EXAMPLE 15 Find the slope of two straight lines that have slope -2 and are tangent to the graph of
DIFFERENTIATION The derivative of can be written as , , , etc
Rules General: Product rule: Quotient rule: Chain rule:
EXAMPLE 16 Differentiate the following: A) B) C) D) E) F) G) H) if , show that
GOOD LUCK