MATH 123 MIDTERM 1 TUTORIAL

Surath GomisSaskatoon Engineering Students’ Society surath.gomis@usask.caOctober 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