MATH109 Lecture Notes - Lecture 12: Difference Quotient

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erinsquirrel903

2 Oct 2018

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University of Waterloo

Department

Mathematics

Course

MATH109

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Brian Leal Fernandes

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MATH109 Lecture 1: Math 109 Note 1

MATH109 Lecture 2: MATH 109 Note 2

MATH109 Lecture Notes - Lecture 12: Difference Quotient

Document Summary

Solve (cid:1876)(cid:1864)(cid:1866)(cid:1876) (cid:1876) (cid:882) so, (cid:1858)(cid:4666)(cid:1876)(cid:4667)=(cid:1876)(cid:1864)(cid:1866)(cid:1876) (cid:1876) (cid:882),(cid:1876)>(cid:882) (cid:1857) (cid:1876) (cid:1864)(cid:1866)(cid:1876) (cid:883) (cid:1858)(cid:4666)(cid:1876)(cid:4667) Math 109 - lecture 12 continuity and derivatives. Discontinuity = where function does not exist = restriction = at (cid:1876)=(cid:882) (cid:887) (cid:883) Goal: to find the slope of a tangent line at a point on a curve. A secant line is a line that intersects a curve at two or more points. A tangent line is a limit of two lines. The slope of a curve at a point is the slope of the tangent at that point. Definition- the derivative of a function f, denoted f" is defined by (cid:1858) =(cid:1864)(cid:1865) (cid:2868)(cid:4666)+ (cid:4667) (cid:4666)(cid:4667) If f"(a) exists, f is said to be differentiable at a. Other notations include: , (cid:1858)(cid:4666)(cid:1876)(cid:4667) , |(cid:1876)= Last notation means, derivative evaluated at x = a . Theorem- if f is differentiable at a , then f is continuous at a.

Related Questions

4. (21 pts) Let g be the function given by ( 6 - X 2 + x if x < 2, x + -2 g(x) = x2 – x-1 if x>2. (a) (6 pts) Use the definition of continuity to determine whether the function g is continuous at 2. Show your work. (b) (3 pts) Evaluate the limit. Justify your answer. Show your work. Write "does not exist" only if the limit does not exist and is not + or -0. lim g(x) = lim -2+ = +00 X 2 + X since the form of the limit is NUM: POS, DENOM: POS (c) (2 pts) Write the equation(s) of all vertical asymptotes of g. Write "none" if appropriate. The result in part (b) implies that the line x= -2 is a vertical asymptote. This is the only vertical asymptote, since the function g is continuous on its domain (d) (2 pts) Determine the largest) intervals of continuity of g. Use interval notation to write your answer. Let g be the function given by 12+ if x < 2, x# -2 2 + x g(x)= x2-x-1 x-1 if x>2. (e) (6 pts) Evaluate each limit. Do not use L'Hôpital's Rule. Write "does not exist" only if the limit does not exist and is not too or -00. Show your work. 6-1 i lim g(x) = lim -00 2 + X = lim -002 + x 0-1 + 1 x -00 x x x -+- 2 +1 0 0 + 11 x2 - x-1 1 ii. lim g(x) = lim x++ V x2 – x-1 X-1 x2 - x-1 X - 1 = lim x +oc x +0 - lim Vam M93001 (f) (2 pts) Write the equation(s) of all horizontal asymptotes of g. Write "none" if appropriate.

1. One of the tables below contains (x,y) values that weregenerated by a linear function. Determine which table, and thenwrite the equation of the linear function represented by the

Table #1 x 2 5 8 11 14 17 20 y 1 3 7 13 21 31 43

Table #2 x 1 2 3 4 5 6 7 y 10 13 18 21 26 29 34

Table #3 x 2 4 6 8 10 12 14 y 1 6 11 16 21 26 31

2. Give an example of each of the following:

A function whosedomain is [2, infinity)

An arithmeticsequence

A system of equationswith no solutions

3. For f(x) =1/x^2-3, find:

f(3)
f(2+h)

4. For f(x) = 1/x-5and g(x) = x^2+2, find:

1. (fog)(x)

2. (gof)(6)

5. For the sequencegiven by a (subscript n) = 4n+5 an=4n+5, answer the following:

Find the first fiveterms

Find the sum of thefirst 25 terms.

Is this an arithmeticsequence? If so, how can you tell? If not, why not?

6. Graph the areabound by y<1/2x+6, x+3y>=12, x>=0, and x<=12

7. For the functiondefined by:

f(x)= {x^2,x<=1}

{2x+1,x>1}

a. evaluate f(0)

b. graph f(x)

8. Solve thefollowing system of equations algebraically. Verify your solutioneither graphically or by using matrices.

3x-y=0

5x+2y=22

9. Solve thefollowing system of equations algebraically. Verify your solutioneither graphically or by using matrices

8x-2y=5

-12x+3y=7

10.Given matricesA, B, and C below, perform the indicatedoperations if possible. If the operation is not possible, explainwhy.

A (3x3) [1st 2, -1,0] [2nd 0, 5, 0.3] [3rd 1, 4, 10]

B (3x3) [1st 5, 0, 2][2nd 1, -3, 9] [3rd2, 0, 4]

C (1x1) [1, 3, 5]

1.3A+B

2. B+C

3. CA

11. Given the tablebelow, evaluate the following:

x	-3	-2	-1	0	1	2	3	4	5
f(x)	10	20	30	40	50	60	70	80	90
g(x)	-1	-2	-3	-4	-5	-6	-7	-8	-9

a.(3f+2g(1) b. (fog)(-1)

12.Express the following function, F(x) as a composition of twofunctions f and g

f(x)= x^2/(x^2+4)

13.You have a coupon for your favorite clothing store for $25 off anypurchase of more than $50. The store is also running a 20%-off saleon its entire inventory. Let x be the original price, f(x)be the price with the $25 coupon applied, and g(x) be the pricewith the 20% discount applied.

a.Write an expression for f(x)

b.Write an expression for g(x)

c.What would the expression (fog)(x) represent?

d.What would the expression (gof)(x) represent?

e.If the store allows you to apply both the 20% discount and the$25-off coupon, does it matter which you apply first? How do youknow?

MATH109 Lecture Notes - Lecture 12: Difference Quotient

MATH109 Lecture 1: Math 109 Note 1

MATH109 Lecture 2: MATH 109 Note 2

MATH109 Lecture Notes - Lecture 12: Difference Quotient

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