Math Test 1

Question 1

Sketch x^2, x^3, logx, e^x, sinx, cosx, tanx, 1/x, sqrt(x), absolute x

Question 2

Which term in a polynomial function will dominate for small x & large x

Answer 2

For small x, smaller degrees of power will dominate.

For large x, larger degrees of power will dominate

Question 3

Sketch & asymptotically show e^x dominates any power function

Answer 3

(e^x, x>k)>x^k

Question 4

Polynomial Power Function

Answer 4

f(x)=Kx^n, K ^ n are constants, can be negative or fraction

Question 5

Polynomial Functions

Answer 5

f(x)=anx^n+an-1x^n-1+…+a2x^2+anx+a0, a & k are constants

Question 6

Sketch f(x)=x^2-x^4, f(x)=3x+x^2, e^x-x^4

Question 7

Hill Function, for small & large x what does it look like

Answer 7

f(x)=(Ax^n)/(B+x^n)

Reason out denominator first, then simplify
For small x, f(x)=Ax^n/B
For large x, f(x)=A

Question 8

Explain with words & graphs what lim(x>a^+/-) f(x)=L

Answer 8

Limit of f(x) as x approaches a is equal to L means f(x) is arbitrarily close to L provided x is sufficiently close to a (but not equal to a)

Limit of f(x) as x approaches a from below is equal to L is arbitrarily close to L provided x is sufficiently close to a and x<a (left hand limit)

Limit of f(x) as x approaches a from above is equal to L is arbitrarily close to L provided x is sufficiently close to a and x>a (right hand limit)

If function does not approach a single number as x approaches a, the limit does not exist (DNE)

Question 9

Explain with words & graphs what lim(x>a^+/-) f(x)=+/- infinity

Answer 9

Vertical asymptotes

Question 10

Explain with words & graphs what lim(x>+/- infinity) f(x)=L

Answer 10

Horizontal asymptotes.

Limit of f(x) is arbitrarily large & +/- provided x is sufficiently close to a (but not equal to a)

Question 11

How to calculate limit

Answer 11

Continuous function’s limits computed by substituting =xaf(x)=f(a)

Not nice (0/0, infinity) function’s limits computed by simplifying into nice function, finding holes, VA & HA asymptotes & substituting

Question 12

How to find features of quotient functions

Answer 12

(x+3)/(x+3)(x-2) Hole: x=-3, Vertical A: x=2 sub in x=1.9, 2.1 to find +/- limits, Horizontal A: n<d y=0, n=d y=coe/coe, n>d y=oblique, sub in x=+/- infinity

Question 13

Identify & explain what a function with a continuous domain is

Answer 13

f(x) is continuous at x=a if xaf(x)=f(x), unbroken, consistent graph, no abrupt changes in value (ex. Polynomials, trig, e^x, log(x))

Discontinuous (ex. piecewise, traffic light)

Question 14

Points of Discontinuity

Answer 14

Jump Discontinuity: piecewise

Removable Discontinuity: hole

Infinity Discontinuity: vertical asymptotes

Question 15

When given a function with paraments, how to select parameter values to make a function continuous

Answer 15

Set equations equal at x=a, if x<k=f(x), xk=g(x), f(k)=g(k), solve for variable

Question 16

Average Rate of Change

Answer 16

Slope of a Secant Line =[f(x1)-f(x0)]/x1-x0

Question 17

Instantaneous Rate of Change

Answer 17

Derivative of f(x) at x0 is f’(x0)=d/dxf(x)=lim(h>0)[f(x+h)-f(x)]/h

Doesn’t exist at corners, sudden changes in slope, jumps or where the limit is infinite number

Question 18

Solve for f’(1), if f(x)=x^2

Question 19

Linear Approximation + Use

Answer 19

Linear Approximation/Tangent Line to f(x) at x=a is L(x)=f(a)+f’(a)(x-a), used to approximate hard values of functions

Question 20

Linear approximation to f(x)=e^x at x=a=0, f’(0)=1

Find e^1/10

Answer 20

L(x)=f’(0)+f’(0)(x-0)=e0+1(x-0) = 1+x

L(1/10)=1+1/10=1.1

Question 21

Euleur’s Number e

Answer 21

lim(h>0) eh-1/h=1

Question 22

Solve for derivative of e^x
& e^f(x)

Answer 22

e^x
e^f(x)*f’(x)

Question 23

Differential Equation + 2 Examples

Answer 23

Differential Equation y’(t)=y(t): a unknown function & its derivatives with a solution that satisfies it

1. y=ex, y’=ex, velocity vs acceleration, compound interest, bacteria growth, population growth
2. y=Cex, y’=Cex, h0Ce(x+h)-Cexh=h0Cexeh-Cexh=h0Cex(eh-1)h=Cex

Question 24

Linearity of Differentiation + Proof

Answer 24

Proof using the limit definition of derivative & use to break down derivatives of sums + constant multiples

Question 25

Power Rule + Proof

Answer 25

Proof using the limit definition of derivative & use for integer exponents

Question 26

Derivative of a function

Question 27

Find derivative of f(x)=3x+x^2

Question 28

Derivative of an exponent (a^x) & tan(x)

Answer 28

d/dx 4^x=4^xlog(4)
d/dx tan(x)=sec2(x)

Question 29

Reciprocal Rule + Proof

Question 30

Product Rule + Proof

Question 31

Find derivative of f(x)=x^2+1/x^2

Question 32

Find derivative of f(x)=x*e^x

Question 33

Quotient Rule + Proof

Question 34

Find derivative of f(x)=(x^3+1)(x^2+2)

Question 35

Graphically show trig f(x)s & their derivatives

Question 36

Addition Formulas

Answer 36

sin(a+b)=sin(a)cos(b)+sin(b)cos(a)
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
limh>0 sin(h)/h=1
limh>0 cos(h)-1/h=0

Question 37

Derivative sinx & cosx proof