Calc 1 exam: trig and limits

sinθ

opp/hyp

cosθ

adj/hyp

tanθ

opp/adj; sin/cos

sin(-θ)

-sinθ (odd)

cos(-θ)

cosθ (even)

sin²θ + cos²θ =

1

tan²θ + 1 =

sec²θ

1+cot²θ =

csc²θ

cscθ

1/sinθ

secθ

1/cosθ

cotθ

1/tanθ

a function f is even when

f(-x) = f(x) for all x. Symmetric about the axis

a function f is odd when

f(-x) = -x. Symmetric about the origin

y = f(x) + k

graph shifts UP by k units

y = f(x) - k

graph shifts DOWN by k units

y = f(x - k)

graph shifts RIGHT by k units

y = (x + k)

graph shifts LEFT by k units

y = cf(x)

STRETCHES graph VERTICALLY

1/cf(x)

COMPRESSES graph VERTICALLY

y = f(cx)

COMPRESSES graph HORIZONTALLY

y = f(x/c)

STRETCHES graph HORIZONTALLY

y = -f(x)

reflects graph across X-AXIS

y = f(-x)

reflects the graph across Y-AXIS

y = |f(x)|

reflects the negative y values across the x axis

informal definition of a limit

Lim x —> a f(x) = L

if the values of f(x) approach L as x approaches a

precise definition of a limit

if for every ϵ > 0 there is a corresponding δ > 0 such that

0 < |x-a| < δ —> |f(x) - L| < ϵ

then

Lim x —> a f(x) = L

Squeeze theorem

f(x) ≤ g(x) ≤ h(x)

sin theorem

limθ —> 0 sinθ/θ = 1

Continuity

a function f is continuous at x = a if

lim (x—>a) f(x) exists

f(a) exists

f(a) = lim(x—>a) f(x)

Intermediate Value Theorem

if f(x) is continuous on [a,b] and N is any number between f(a) and f(b), then there exists a number c in (a,b) such that f( c ) = N

Horizontal asymptotes

y = L is a horizontal asymptote if

Lim (x—>∞) f(x) = L

Lim (x—>-∞) f(x) = L

Vertical Asymptotes

x = a is a vertical asymptote if

Lim(x—>a-) f(x) = ±∞

Lim(x—>a+) f(x) = ±∞