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Question 1

What is it called when the left limit doesn’t equal the right limit?

Answer 1

A divergent limit

Question 2

How do you prove a limit is continuous?

Answer 2

• Show that the limit at x=a exists, meaning the left and right limits are equal
• And that the limit is equal to the function at value x=a (and is finite

Question 3

How do you convert to cartesian form from polar?

Answer 3

x=rcos0
y=rsin0

Question 4

How do you convert from cartesian to polar spherical?

Answer 4

r=sqrt(x2+y2+z2)
O/=tan-1(y/x)
0=cos-1(z/r)

Question 5

How do you convert from polar spherical to cartesian?

Answer 5

x=rsin(0)cos(O/)
y=rsin0sin0/
z=rcos0

Question 6

How do you convert from polar cylindrical to cartesian?

Answer 6

x=rcos0
y=rsin0
z=z

Question 7

What is limx–>0(sinx/x) equal to?

Answer 7

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Question 8

Differentiate ln(3x)

Answer 8

(1/3x)*3=1/x

Question 9

What is the differentiation rule for fractions?

Answer 9

(vdu/dx - udv/dx)/v2

Question 10

Differentiate tan(x)

Answer 10

sec^2(x)

Question 11

How do you classify stationary points?

Answer 11

• Find the second derivative, and sub in the stat pt
• If second derivative is smaller than zero, then local max at x
• If it is bigger than zero, local min
• If equal to zero, need higher order derivative test

Question 12

What are the derivatives of sin^2(ax) and cos^2(ax)?

Answer 12

sin^2(ax) - 2asin(ax)cos(ax) or asin(2ax)
cos^2(ax) - -2acos(ax)sin(ax) or -asin(2ax)

Question 13

What are sin(2x), cos(2x) and tan(2x) equal to in double angle formulas?

Answer 13

sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x)-sin^2(x) = 2cos^2(x)-1 = 1-sin^2(x)
tan(2x) = 2tan(x)/1-tan^2(x)

Question 14

What are the 4 main steps for curve sketching?

Answer 14

• Find whether the function is odd or even (even: f(-x)=f(x), odd: f(-x)=-f(x)
• Find intercepts
• Find asymptotes (lim x–>+-infinity for horizontal, where limit goes to +- infinity for vertical
• Find stationary points and their nature with second derivative