Unit 2

Alternate Derivative Form (IROC)

*f(a) is plugging in the actual value a into the equation, while f(x) is only plugging in the equation

AROC

*slope of secant line over an interval [a, b] f(x) - f(a) / x - a OR f(a + h) - f(a) / (a + h) - a

IROC

*slope of tangent line at one point/derivative

difference quotient

*use this if you are only given one point and are asked to find the derivative at that point

derivative of cosx

-sinx

lim h->0 cosh-1/h

0

a relative min occurs on the function if f'(x) = ______ and the derivatives changes from _______ to ________

0 negative positive

a relative max occurs on the function if f'(x) = ______ and the derivatives changes from _______ to ________

0 positive negative

At a turning point or max/min, the tangent is...

0 or horizontal

At a point of inflection (POI) on f(x), concavity changes, the tangent on f'(x) is...

0 or horizontal - this is where a min/mix occurs on f'(x)

lim h->0 sinh/h

1

differentiability and continuity

1. If a function is differentiable at x = c , then it is continuous at x = c. So, differentiability implies continuity. 2. It is possible for a function to be continuous at x = c and not be differentiable at x = c. Continuity does not imply differentiability.

procedure to sketch an f'(x) "slope graph" from f(x) = y

1. describe how the slope changes from left to right along the curve, Notice the intervals where the slope is positive, negative and zero. 2. Write down the changes as they occur from left to right on a number line. Notice where the x-intercepts occur. Pay attention to where the function is above the x-axis (positive) and where it is below the x-axis (negative).

derivative of y=lnx

1/x, x>0

differentiability at a point

A function y = f(x) is differentiable at x = c if and only if the limit as x approaches the derivative of the point from the left and right are equal. f(x) - f(c) / x - c

Defending your answers. If you are given that y = f(x) is a differentiable function at x = 3, explain why each statement below is true. A. f(3) exists B. lim x →3 f(x) exists C. lim x → 3 f(x) − f(3) / x - 3 exists D. lim h → 0 f(3+h) − f(3) / h exists

A. If f is differentiable at x = 3, then f is continuous at x = 3. If f is continuous at x = 3, then f(3) must exist. B. If f is differentiable at x = 3, then f is continuous at 3. If f is continuous at x = 3, then the lim x → 3 f(x) exists. C. This is the alternate definition of the derivative. It states that the derivative of f exists at x = 3. If f is differentiable at x = 3, then the derivative exists at x = 3. D. This is the limit definition of the derivative, and says that the derivative of f at x = 3 exists. If f is differentiable at x = 3, then the derivative exists at x = 3.

at a "sharp corner" or "cusp," the tangent...

DNE

when the curve is concave down, the tangent line lies...

above the curve (except at point of tangency)

When f is increasing, f' is...

above the x-axis

when the curve is concave up, the tangent line lies...

below the curve (except at point of tangency)

When f is decreasing, f' is...

below the x-axis

derivative of sinx

cosx

cos(x + h)

cosx cosh - sinx sinh

derivative of a constant function

d/dx [c] = 0 if c is any real number

sum rule

d/dx [f(x)+g(x)] = f'(x)+g'(x)

difference rule

d/dx [f(x)-g(x)] = f'(x)-g'(x)

single variable rule

d/dx [x] = 1

power rule

d/dx[x^n]=nx^(n-1)

When f is concave down, f' is...

decreasing

if f'(x) < 0, then the graph of the function is...

decreasing at x = a

if f(x) is NOT continuous at x = c, then f(x) is NOT...

differentiable at x = c

derivative of y=e^x

e^x (equals itself)

When the graph of f(x) changes concavity at x = a, an...

extrema is created on f'(x). Wherever extrema occurs on f'(x), that is a x-intercept on the graph of f(x)

definition of a derivative

f'(x) = limh→0 [f(x+h) - f(x)] / h if f'(x) exists, f is differentiable at x

the IROC formula gives you the derivative, or the equation of the tangent line. you can use this to...

find the slope of the tangent at a single point by plugging in x values.

When f is concave up, f' is...

increasing

if f'(x) > 0, then the graph of the function is...

increasing at x = a

vertical tangent line

occurs when lim h -> 0 f(a + h) - f(a) / h = negative or positive infinity f is continuous at a

derivative of tanx

sec²x

to see which values make a function differentiable (piecewise function)...

set the expressions with a common value (x) equal to each other and plug in that x value to see if it makes the two expressions equal each other. If they equal each other, the x value makes the function differentiable. if they don't, the x value does not make the function differentiable.

sin(x + h)

sinx cosh + cosx sinh

slope of a curve at a point

slope of the tangent line, derivative, or IROC

when we say a function is differentiable, we mean that...

the derivative exists or you are able to take the derivative. that in order for a function to be differentiable at a point, the graph must be continuous and smooth at the point.

general power rule

the derivative of ax^n, where a and n are real numbers, is (a ⋅ n)x^n-1

at a point on a linear function, the tangent is...

the same slope as the line