Sohalski's Super Steller Semester Review

Trig Identities

100

sin²x+cos²x=?

What is 1?

200

sin(2x)=?

What is 2sinxcosx?

300

cot(π/3)=?

What is (√3)/3?

400

Is sec even or odd?

What is odd?

500

cos(2x)=?

Name two

What is...?

cos²x - sin²x

2cos²x - 1

1 - 2sin²x

Theorems

100

If f(2)= 4 and f(3)=9 and f(x) is continuous on all intervals, this theorem guarantees that there exists a point c where f(c)=8 on the interval (2,3)

What is IVT?

200

f(x) is continuous on [2,6]

Is f(x) differentiable on the interval (2,6)?

What is no?

continuity does not imply it is differentiable 

300

If f(x)=-x²+6x-6  for [1,5] and f(1)=5 and f is continuous on [1,5] and differentiable on [1,5], this theorem guarantees at least one value x=c such that f'(c)=0

What is Rolles Theorem?

400

By what law is this possible:

∫₂⁴ f(x)dx = F(4)-F(2)

What is the Fundamental Theorem of Calculus?

500

lim_x→∞ sinx/x = 0  because

-1<sinx<1

lim_x→∞ 1/x = 0

lim_x→∞ -1/x = 0

The work above demonstrates this theorem

What is Squeeze Theorem?

Derivative Rules

100

f(x) = cosx

f'(x) = ?

What is -sinx?

200

f(x) = secx

f'(x) = ?

What is secxtanx?

300

h(x)=f(g(x))

h'(x)= ?

What is f'(g(x)) * g'(x)?

400

f(x) = arccscx

f'(x) = ?

What is  -1/(lxl*√(x²-1))?

500

f(x) = log_ax

f'(x) = ?

What is (1/xlna)?

Integral Rules

100

∫(1/x) dx = ?

What is  ln lxl +C ?

200

∫csc²(x) dx

cot(x)+C

300

∫a^x dx

(1/lna)a^x +C

400

∫(du/u√(u²-a²))

(1/a)arcsec(|u|/a) +C

500

∫sec(x) dx=

ln|sec(x)+ tan(x)| + C

Potpourri

100

The particle's position is given by the function s(t), where t is time in time, what does the equation s'(3) indicate?

The velocity of the particle at time 3

200

Let h(x)=x³+6x²+2

What is the absolute minimum value of h over the closed interval -6 ≥x ≥2

Justify your answer.

The Absolute Minimum value  on [-6,2] is 2

h(-6) is an endpoint, h(-6)= 2

h(0) is a critical number b/c h'(0)= 0

h(0)= 2

h(2) is an endpoint, h(2)= 34

300

Let h be twice differntiable function, and let h(-2)=2, h'(-2)=0 and h"(-2)=-2

What occurs in the graph of h at the point (-2,2)

(-2,2) is a relative max b/c h'(-2)= 0 and h"(-2)<0

400

The cumulative cost of purchasing and maintaining Julia's computer is increasing at a rate of r(t), dollars per year (where t is the time in years). At t=1, Julia had spent a total of $420, on her computer.

What does ∫⁵₁ r(t) dt=570 mean?

By the end of the fifth year, Julia had spent a total of $570,purchasing and maintaining her computer.

500

write the integral represented by the Riemann sum

lim_n→∞ Σⁿ_{i=1  (e}^3i/n)(3/n)

∫₀³ e^x dx