MAT 132 MIDTERM FORMULAS

What does the derivative for a parametric curve equal?

(dy/dt)/(dx/dt)

What is the point slope formula?

y -y1 = m(x-x1)

What is the area of a parametric function?

The integral of y(t)x’(t)

What is the arc length of a parametric function?

The integral of sqr(dy/dt)² +(dx/dt)²

How do you covert polar coordinates to cartesian?

rcos(theta) = x and rsin(theta) = y

How do you convert rectangular coordinates to polar?

r = Sqr(x² +y²) and tan^-1 = y/x

How do you calculate the arc length of a polar curve?

L = the integral of sqr(r² + dr/d(theta)

What is the formula for an arithmetic sequence?

dn = d1 + (n-1)d

What is the formula for a geometric series?

an = ar^n-1

What is the formula for the SUM of a geometric series for a FINITE sum?

Sn = a(1-r^n)/1-r.

What is the formula for the INFINITE SUM of a geometric series

S = a/1-r. Use this when the absolute value of r is less than one.

What is the formula for the second derivative of a parametric curve?

(d/dx) (dy/dt)/dx/dt)

What is the formula for the sum of an infinite arithmetic series?

Sn = n(n+1)

Describe the Divergence Test

If the limit of a sequence is not equal to zero, then the infinite series diverges. If the limit of sequence is equal to zero, then we don’t know if the series converges or diverges.

For a geometric series, how do you know if it’ll converge or diverge?

If the common ratio r<1, then the geometric series will converge. Then you will use the

What criteria must be met in order for the integral test to be used?

an must be positive, and continuous and decreases when x is greater than or equal to 1

What is the p series test?

For any series in the form 1/n^p, if the value of p is greater than 1 it converges, and if it is less than or equal to 1, it diverges

What is arctan bounded by?

-pi/2 and pi/2

What is the formula for the area of a polar function?

A = the integral of 1/2r² or A = the integral of 1/2f(theta)

What is the ratio test?

Take the limit as n approaches infinity of the absolute value of an+1/an. If the limit is less than 1 it converges, If it is greater than or equal to 1 or is infinity, it diverges. If it is equal to 1, the test is inconclusive.

What is the root test?

Take the limit as n approaches infinity of nroot of the absolute value of an. If the limit is less than 1, it converges. if the limit is greater than or equal to 1, it diverges. If it is equal to 1 it is inconclusive.

What is the Direct Comparison Test?

If the big series coverges, so does the smaller series. If the smaller series diverges, the big series does as well.

What is the limit comparison test?

The limit as n approaches infinity of an/bn converges, both series converge and vice versa.

What is the alternating series test?

In the form (-1)^n (an), it must past the divergence test. Then you need to show that the sequence decreases.

What is the absolute value test?

If the absolute value of an converges, then the original series is absolutely convergent. If the absolute value of an diverges but the original ser