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- Calculus
- Calculus 1610
- Ghimire
- Final Review

Andrada N.

y= f(x)+c

Vertical shift of the graph of y= f(x) a distance c units upward

y=f(x)-c

Vertical shift of the graph of y= f(x) a distance c units downward

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y=f(x-c)

Horizontal shift of the graph of y= f(x) a distance c units to the right

y=f(x+c)

Horizontal shift of the graph of y= f(x) a distance c units to the left

y=f(-x)

Horizontal reflection of the graph of y= f(x) across the y-axis

y=-f(x)

Vertical reflection of the graph of y= f(x) across the x-axis

y=f(^{x}/_{c})

Horizontal stretching of the graph y=f(x) by a factor c

y=f(cx)

Horizontal shrinking of the graph y=f(x) by a factor c

y=c(f(x))

Vertical stretching of the graph y=f(x) by a factor c

y= ^{1}/_{C}•f(x)

Vertical shrinking of the graph y=f(x) by a factor c

a^{y} = x

Rearrange equation in logarithmic form.

log_{a}x=y

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log_{a}1 =

o

a^{0} = 1

log_{a}a =

1

a^{1} = a

e^{y} = x

Rearrange equation in logarithmic form.

y = log_{e}x

lnx is the same thing as...

log_{e}x

log_{a}(xy) =

log_{a}x + log_{a}y

log_{a}(x/y) =

log_{a}x - log_{a}y

log_{a}x^{r} =

r log_{a}x

(where r = real number)

lne^{x} =

x

(ln and e cancel each other out)

e^{lnx} =

x

(ln and e cancel each other out)

Conditions for Continuity

1. f(a) exists

2. lim as x tends to a exists

3. lim as x tends to a = f(a)

What 6 functions are continuous at every number in their domain?

1. Polynomial

2. Rational

3. Root

4. Exponential

5. Logarithmic

6. Trigonometric

d/dx(sinx) =

cosx

d/dx(cosx) =

-sinx

d/dx(tanx) =

sec^{2}x

d/dx(cscx) =

-cscx•cotx

d/dx(secx) =

secx•tanx

d/dx(cotx) =

-csc²x

Implicit Differentiation

A relation **F(x,y) = 0** is said to **define the function y = f(x) implicitly** if, for **x** in the domain of **f**, **F(x,f(x)) = 0. **Replace the *y* with y(x) [y as a function of x]* *and then take the derivative. (Differentiate both sides of the equation with respect to "x", and then solve the equation for dy/dx).

If s=f(t), position function of a particle that is moving in a straight line, then...

velocity v(t) = ds/dt = s¹

Acceleration a(t) =

a(t) = dv/dt = ds¹/dt = s¹¹

or

a(t) = v¹ =ds¹/dt = s¹¹

L'Hospital's Rule

Suppose that f and g are differentiable and g'(x)≠0 on an open interval I that contains a. Suppose that f(x)/g(x) gives and indeterminate form of type 0/0 or ∞/∞. ^{x→a }g(x) ^{ } ^{x→a }g'(x)

Then lim __f(x)__ = lim __f'(x)__

if the limit on the right side exists (or is ∞ or -∞)

The definition of a derivative

f'(x)= lim h-->0 (f(x+h)-f(x))/h

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