Calculus 1 Exam 1 Terms

y = f(x) + c

Moves up

y = f(x) - c

Moves down

y = f(x+c)

Moves left

y = f(x-c)

Moves right

y = c * f(x)

Stretches vertically

y = (1/c) * f(x)

Shrinks vertically

y = f(c*x)

Shrinks horizontally

y = f(x/c)

Stretches horizontally

y = -f(x)

Flip over x-axis

y = f(-x)

Flip over y-axis

lim [f(x) - g(x)] = …?

lim f(x) - lim g(x)

lim [f(x) + g(x)] = …?

lim f(x) + lim g(x)

sin²x + cos²x =

1

1 + tan²x =

sec²x

1 + cot²x =

csc²x

sin (a+-b) =

sin(a)cos(b)+-cos(a)sin(b)

cos(a+-b) =

cos(a)cos(b)-+sin(a)sin(b)

sin(2x) =

2sin(x)cos(x)

cos(2x) =

cos²x-sin²x

lim (sin(x) / (x)) =

1

lim (tan(x) / (x)) =

1

lim (1-cos(x) / (x)) =

1

lim (1-cos(x) / (x²)) =

1/2

lim (sin(ax) / (x)) =

a

lim (tan(ax) / (x)) =

a

What is the difference quotient

(f(a+h)-f(a))/h

As x→2-, rewrite |x-2| as…?

-(x-2)

Square roots __ 0, logs __ 0, absolute value __ 0

> or equal to; >; > or equal to

For absolute value and continuity limits, you must…?

Check left and right

f(a) must be __________ and the __________ must exist in order for continuity to be possible

Defined; exist

Intermediate Value Theorem: If a function is continuous, it must…? If a function is continuous and it is positive and becomes negative, it must…?

Pass through all values in between; Have an x-intercept

How do you solve piecewise discontinuity problems?

Check the left and right of joining / problem points. If they are all equal, the function is continuous.

For infinite limit problems, when given lim x→a number, what must you find? What should your answer be?

Signs of the numerator and denominator; -∞, ∞, or DNE

How do you find vertical asymptotes?

Find where (only) the denominator goes to zero / und

If lim x→∞, what is 1/x?

0

e^x = ?

1

If you have e^x and you’re solving for x, what should you do?

Multiply by ln()

v(t) = ?; a(t) = ?

s’(t); v’(t)

What is the equation for average velocity / instantaneous velocity? What is the interval variables?

lim h→0 ( f(x+h) - f(x) ) / h; Where the interval is [a, a+h]

Rate of change is the same as __________ __________

Average velocity

d/dx (sin x)

cos x

d/dx (cos x)

-sin x

d/dx (tan x)

sec²x

d/dx (-csc x)

-csc x cot x

d/dx (sec x)

sec x tan x

d/dx (cot x)

-csc²x

tan x =

sinx/cosx

log_a1 = ?

0

log_aa = ?

1

log_aa^x = ?

x

a^log_ax = ?

x

ln 1 = ?

0

ln e = ?

1

ln e^x = ?

x

e^lnx = ?

x

log 50 + log 20 = ? = ?

log 1000; 3

What is the equation for the Inverse Function Theorem

(f^-1)’(a) = 1 / (f’(f^-1(a)))

What are the (4) steps to finding the derivative of an inverse function using the Inverse Function Theorem?

1) Find f’(x)
2) Find f^-1(a) = x, where f(x) = a

3) Plug that number into f’(x)

4) Take reciprocal

What is the equation for L(x) Linearization

L(x) = f(a) + f’(a)(x-a)