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Calculus Test 1 > Study > Flashcards

Study Flashcards

1
Q

Why is 3ln(1/2) < 2ln(1/2)?

A 
3ln(1/2) = 3ln(2^-1)=-3ln(2)
and 2ln(1/2)=2ln(2^-1)=-2ln(2).

-3ln(2) < -2ln(2)

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2
Q

If a function has a curved top what is the interval?

A

Closed

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3
Q

An interval is closed when asked to find where a function is increasing/decreasing when

A

The top is rounded

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4
Q

Why is sin^-1(sin 4π/3) = -π/3?

A

sin(π-θ)= sin, so:

sin(π-4π/3)=sin(-π/3) which is in the range of sin^-1.

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5
Q

sin(π-θ)= sin, so:

sin(π-4π/3)=sin(-π/3) which is in the range of sin^-1.

A

sin^-1(sin 4π/3) = -π/3

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6
Q

What does it mean to find a domain where f is one-to-one?

A

Find where it is defined

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7
Q

Simplify with exponent rules:

√a √b/ √ab

A

a^1/6 / b^1/12

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8
Q

Find domain:

h(x)=1/ ^4√x^2 -5x

A

(-∞, 0) U (5, ∞) because x(x-5)>0 is equal to x<0 and x>5

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9
Q

Find domain:

f(x)= √16-x^4 / x^2 -1

A

[-2,-1)U(-1,1)U(1,2] because x cannot equal +-1 and x<=+-2 which equals -2<=x<=2

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10
Q

If a function is always decreasing, never zero, and f(-2)=1 and f(1)=2, what is the graph?

A

A piecewise function.

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11
Q

What does cos(-x) equal?

A

cos(x)

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12
Q

What does sin(-x) equal?

A

-sin(x)

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13
Q

Equals -sin(x)

A

sin(-x)

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14
Q

Simplify:

2x(4-x)^-1/2 - 3√4-x = 0

A

x=12/5 because you put (4-x)^-1/2 under 2x and ass 3√4-x to right side. Then solve for x.

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15
Q

How is this absolute value function solved?:

3|x-4|=10

A

|x-4|=10/3

Then x-4=10/3 and x-4=-10/3

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16
Q

Simplify:

3x^3/2•y^3 / x^2•y^-1/2

A

9y^7 / x because you multiply all exponents by 2 first. Then any negative exponents are flipped and solve.

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17
Q

How is this simplified?

√200 - √32

A

Split into one non-squarable and squarable:

√2 • √100 - √2 • √16

√2 • 10 - √2 • 4

6√2

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18
Q

Evaluate:

y/x - x/y / 1/y - 1/x

A
  • (y+x) because (y+x)(y-x) / x-y =

- (y+x)(x-y) / x-y

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19
Q

How is this evaluated?:

16^-3/4

A

Equals 1/(16^1/4)^3

1/2^3

1/8

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20
Q

How do you prove:

√x^2 = x

A

√(-2)^2 = -2 so false

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21
Q

How do you prove:

a^2+b^2=c^2

A

(1)^2+(2)^2=(3)^2 so false

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22
Q

How do you prove:

√xy

A

Equals √x•√y

x^1/2 • y^1/2

(xy)^1/2 = √xy so true

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23
Q

If the half-life is 4 days, what is the mass after 16 days?

A

(1/2)^t/4 where t is days and 4 is half-life

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24
Q

Difference between rational and algebraic equations

A

Rational has polynomial on numerator AND denominator

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25
Q

Rational has polynomial on

A

numerator AND denominator

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26
Q

Evaluate this log:

log(5)(200/8)

A

log(5)(25)

so 2

27
Q

Solve for x:

2^x = 5^3x+7

A

x=7ln(5) / ln(2) - 3ln(5) because you take exponent out and distribute.

28
Q

If you have domain and need range, what is done?

A

Test values of domain for highest and lowest values of range.

29
Q

Test values of domain for highest and lowest values of range.

A

If you have domain and need range

30
Q

Why is cos^-1(-1)= π?

A

Because cosine hits -1 every π

31
Q

Because cosine hits -1 every π, this means the inverse would be.

A

cos^-1(-1)= π

32
Q

What is the domain of ln(x)<0?

A

3^ln(x)

33
Q

Definition of one-to-one funct

A

Function that never takes on the same value twice

34
Q

Function that never takes on the same value twice

A

One-to-one

35
Q

Find exponential function given two points:

1,12) and (3,108

A

12=C(b^1)=Cb
108=C(b^3)=Cb^3

C=12/b (solve for C)

108= 12/b•(b^3)=12b^2 (plug in C in second)

b=3 (solve for b)

C=12/3 = 4 so,

y=4(3)^x

36
Q

How is number e defined?

A

The value so that the slope of the tangent like to y=e^x at (0,1) is exactly 1.

37
Q

The value so that the slope of the tangent like to y=e^x at (0,1) is exactly 1.

A

e

38
Q

How is this divided?:

9^3 / 3^-8

A

Flip 1/9^3 / 1/3^8

1/9^3 • 3^8

3^8 / 9^3

3^8 / (3^2)^3

3^8 / 3^6

3^2

39
Q

The graph of y=1/3x + b is shrunk by a factor of

A

3 not 1/3

40
Q

Is f(s+t)=f(s)+f(t)?

A

No because (f+f)(x)=f(x)+f(x)

41
Q

Suppose g is an even function and let h=f o g. Is h always an even function?

A

Yes because if g = x every time being even, then so is f o g

42
Q

Volume of sphere

A

4/3πr^3

43
Q

4/3πr^3

A

Volume of sphere

44
Q

Graph the function y=|x^2 -9|

A

Same as regular x^2 graph but x’s from -3 to 3 are reflected over x-axis

45
Q

Equation for two families if functions if one family’s is y=5x+b and the other’s is y=m(x-5)+1.

A

5x+b=m(x-1)+1 and solve

46
Q

If a box has dimensions 18in. by 30in and gets equal squares of side x at each corner folded up, express the new volume as a function of x.

A

V=lwh so,

V=(30-2x)(18-2x)x

47
Q

If the sides of a rectangle box, (not the bases), are the height (h) and the length (2w), and also the width (w), what is the cost of all the sides?

A

2[2(wh) + 2(2wh)] since the area for 2 sides is 2(wh) total, and the other two sides is 2(2wh) total.

48
Q

If the material for the base of a rectangular box costs $8 per square meter, what is the total cost if w is the width and 2w is the length?

A

A=lw so,

(2w)(w)=2w^2

Multiply 8 by this number for the cost:

8(2w^2)

49
Q

In general, if f(x)=g(x) when x cannot equal a, then

A 
lim f(x) = lim g(x)
x—>a       x—>a
50
Q

Substituting limit points directly into functions only works for

A

Polynomials and rationals

51
Q

Only works for polynomials and rationals

A

Substituting limit point directly into a function

52
Q

Domain of tan function

A

nπ/2

53
Q

nπ/2

A

Domain of tangent function

54
Q

How is cot graphed?

A

Tangent but backwards

55
Q

Tangent but backwards on graph

A

Cot

56
Q

Domain and range of cos^-1

A

R: [0, π]
D: [-1,1]

57
Q

Domain and range of sin^-1

A

D: [-1,1]
R: [-2π, 2π]

58
Q

When reflection about a y value, what is done?

A

Add negative at base and add twice the y-value

59
Q

Add negative at base and add twice the y-value

A

When reflection about a y value

60
Q

When reflecting about x-values what is done?

A

Add negative at x and add twice the x-value

61
Q

Add negative at x and add twice the x-value

A

When reflecting about x-values

62
Q

Order of transformations

A 
H-shift
H-stretch/shrink
Reflect y-axis
Vert-stretch/shrink
Reflect x-axis
Vert-shift
63
Q

To find x-values of log function what is done?

A

Set praenthesis equal to 0, 1, and the base and solve

64
Q

Set praenthesis equal to 0, 1, and the base and solve

A

To find x-values of log function

Calculus Test 1 (1 decks)

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