Unit 1

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line. y = x² + 6x

(-3, -9)

Rewrite d/dx[7/4x⁵]

(-35/4)(x⁻⁶) or -35/4x⁶

Find the limit. lim (-x²-x) x → -2

(-x²-x) -(-2²) - (-2) -(4) - (-2) -4 + 2 -2

Determine a point (x,y) where f(x) = 4x² + 2x has a horizontal tangent line.

(-¼, -¼)

Determine the intervals over which the function is continuous. (t-2)(t+3) -------- = v(t) (t²-4)

(-∞, -2)∪(2, ∞)

Determine the intervals over which the function is continuous. f(x) = 3 if x ≤ 2 x²-1 if 2 < x ≤ 9

(-∞, 9)

Determine the intervals over which the function is continuous. y = 7x² - x

(-∞, ∞)

Determine intervals of continuity. g(x) = (x²-8x+12)/(x²-4)

(-∞,-2)∪(-2,∞)

Find the limit using direct substitution. lim (5x+2) / (3−x) x → −3

(5x+2) / (3−x) (5(-3)+2) / (3−(-3)) (-15+2) / (3-(-3)) -13 / 6

Find the limit (if it exists). lim (6x² − x − 7) / (x + 1) x → −1

(6x² − x − 7) / (x + 1) (x+1)(6x-7) / (x+1) 6x-7 6(-1)-7 -6-7 -13

Find f'(x) f(x) = (8x² - 7x + 1) / x

(8x² - 1) / x²

Find the limit (if it exists). lim (t² + 4t − 12) / (t² − 4) x → 2

(t² + 4t − 12) / (t² − 4) (x-2)(x+6) / (x-2)(x+2) (x+6) / (x+2) (2 + 6) / (2 + 2) 8 / 4 2

lim f(x) = (x - 2/x² - 3x + 2) x → 2

(x - 2/x² - 3x + 2) (x - 2) / (x - 2)(x - 1) 1 / (x - 1) 1 / (2 - 1) 1 / 1 1

lim f(x) = (x² - 3x + 1) x → 2

(x² - 3x + 1) 2² - 3(2) + 1 4 - 6 + 1 -1

How do you find the slope of a secant line?

(y₂ - y₁) / (x₂ - x₁)

lim f(x) (√x+4 - 2)/x x → 5

(√x+4 - 2) /x √5+4 - 2 / 5 3 - 2 / 5 1/5

Find the limit using direct substitution. lim (√x-3 - 2) / x x → 12

(√x-3 - 2) / x (√12-3 - 2) / 12 (√9 - 2) / 12 (3 - 2) / 12 1 / 12

Use the limit definition to find the slope of the tangent line to the graph of f at the given point. f(x) = 25 − x² (5, 0)

-10

Find the limit using direct substitution. lim (−x² + x − 4) x → 4

-16

Find the limit using direct substitution. lim (9x − 2) x → 0

-2

Find f'(x) for f(x) = x⁻²

-2x⁻³ or -2/x³

Determine the average rate of change of the function over the interval [-3, 1]. g(x) = x² - 5x

-7

Find f'(x). 4 -- = f(x) x²

-8x⁻³

Derivative

-A formula that gives the slope of the tangent line -Notated by: y, y', f'(x), dy/dx

Continuity at a Boundary or Endpoint

-Continuous in an interval, not the whole way through -Can only approach a point from one direction -Limit of one side must match the solid point of the function

How do you find a limit if no graph is given?

-Direct substitution -Factoring and cancelling -Envision a graph for a piecewise function

How do you find the limit via a peacewise function without a graph?

-Draw the "seam" -Plug in the seam value(s) into the functions -Determine if the limit exists f(x) = x²+1 x≥1 4 x<1 = 1² + 1 = 2 = 4 = 4 The limit does not exist

How do you find the limit of f(x)?

-Look at the direction the line is moving -See which Y value it is apporaching

Average Rate of Change

-Rate of change overall -Slope

Consider the following. f(x) = 8 − 2x − x² 1. Describe the interval(s) on which the function is continuous. 2. Identify any discontinuities. 3. If the function has any discontinuities, identify the conditions of continuity that are not satisfied. a. There is a discontinuity at x = c where f(c) is not defined. b. There is a discontinuity at x = c where lim f(x) ≠ f(c) x→c c. There is a discontinuity at x = c where lim f(x) does not exist. x→c d. There are no discontinuities; f(x) is continuous.

1. (-∞,∞) 2. DNE 3. d. There are no discontinuities; f(x) is continuous.

Find the limit. lim (x-5) x→5 (x²-25)

1/10

Use the limit definition to find the derivative. f(x) = 5x² - 2x + 3

10x - 2

Find a function for the marginal profit. P(x) = 5x² - √x

10x - ½x⁻¹/²

Find f'(x). f(x) = 3x⁴ - 2x³ + 7x² - 9x + 11

12x³ - 6x² + 14x -9

Determine the limit. lim (2x³ - 1) x→2

15

Rewrite y = x∧²/³

2/3x∧⁻¹/³ or -2/3x∧¹/³ or -2/3³√-x

Find the limit. lim (4x² - 5) x→2

27

Find f'(x) f(x) = x² + 4x + 5/x

2x + 4 - 5x⁻²

Use the limit definition of the derivative to find f'(x). f(x) = x² - 2x + 4

2x - 2

Find f'(x) f(x) = x² -9x - 6x⁻² + 5x⁻³

2x - 9 + 12⁻³ - 15x⁻⁴

What is the equivalent of 2x/x⁻⁷?

2x⁸

Find f'(x) f(x) = (x³ + 2x)(x - 1)

4x³ - 3x² + 4x -2

Use the limit definition of the derivative to find f'(x). f(x) = 3x² - x + 2

6x - 1

Using f'(x) = 8x - 7, determine the instantaneous rate of change of f(x) when x = 10.

73

Find the limit. lim 8 x→6

8

Find f'(x) f(x) = 4x² + 3x + 5/x

8x + 3 - 5x⁻²

Limit Definition of a Derivative

A formula to algebraically find the derivative

Secant

A line that cuts through the curve of a graph

Power Rule

A shorter way to find the derivative x² → 2x 1. Multiply the exponent value by the coefficient (2)1x = 2x 2. Subtract 1 from the exponent value x²⁻¹ = x¹

What is a limit?

A way of getting "f(x)" as close to the limit (L) by getting "x" close to a point (c)

What does f(x) mean?

Function of f?

Determine whether the function is continuous on the entire real number line. Explain your reasoning. f(x) = (x² − 25)³ a. The function is not continuous because the function is not a polynomial. b. The function is not continuous because the function is not defined at x = 25. c. The function is not continuous because the function is not defined at x = ±5. d. The function is continuous because the function is a polynomial. e. The function is continuous because the function's domain is the entire real line.

d. The function is continuous because the function is a polynomial.

Determine whether the function is continuous on the entire real line. Explain your reasoning. g(x) = (x² − 8x + 15) / (x² − 16) a. The function is not continuous because the function is not a polynomial. b. The function is continuous because the function is a polynomial. c. The function is continuous because the rational function's domain is the entire real line. d. The function is not continuous because the rational function is not defined at x = ±4. e. The function is not continuous because the rational function is not defined at x = 16.

d. The function is not continuous because the rational function is not defined at x = ±4.

Find the derivative of the function. f(x) = 1 / x⁵

f'(x) = -5 / x⁶

Use the limit definition to find the derivative of the function. f(x) = 8x² + 9x

f'(x) = 16x + 9

Use the limit definition to find the derivative of the function. f(x) = t² - 7

f'(x) = 2t

Find the derivative of the function. f(x) = x² + 7x³

f'(x) = 2x + 21x²

Use the limit definition to find the derivative of the function. f(x) = 3t² /8

f'(x) = 3t / 4

Use the limit definition to find the derivative of the function. f(x) = 5x

f'(x) = 5

Find the derivative of the function. f(x) = 14x∧(1/2)

f'(x) = 7 / √x

Determine continuity. f(x) = x²-4 x≤0 3x+1 x>0

f(x) = x²-4 x≤0 3x+1 x>0 = 0²-4 = -4 = 3(0)+1 = 4 = (-∞,0)∪(0,∞)

Find the derivative of the function. h(x) = 2x⁹

h'(x) = 18x⁸

What is the equivalent of ∧m√x∧n?

x∧n/m

What is the equivalent of ²√x³?

x∧³/²

Find the derivative of the function. y = 7x³ − x² + 2x − 6

y = 21x² - 2x + 2

Find the equation of the line tangent to f(x) at the point (-3,-9). f(x) = -x²

y = 6x + 9

Determine the equation of the line that is tangent to the function f(x) at the point (1,2). f(x) = -x² + 3x

y = x + 1

Find f '(x). f(x) = x∧¼ − 4

¼x∧-¾

What symbol may be used in place of "h" in the limit definition?

∆E

Find the limit. lim √x x→81

√x √81 9

If the left and right limits do not match, the line (is/is not) continuous.

Is not

If there are any variables in the denominator of a fraction then the algebraic expression (is/isn't) a polynomial.

Isn't

Which direction is the limit approaching from if x → 3⁻?

Left

Which direction is the limit approaching from if x< or x≤?

Left

Sum Property and Difference Rule of Limits

Limits can be added and subtracted like constants

Constant Multiple and Quotient Property of Limits

Limits can be multiplied and divided like constants

What can you do if you need to change the signs during factoring?

Multiply by -1

When figuring out the limit of a combined function, if one limit does not exist, the function's limit can (never/sometimes/always) exist.

Never

If both the left and right limits exist, but the point is not solid, or a solid point is present off of the line, is the function continuous?

No

How do you find y if it is not provided?

Plug x into the original equation

Which direction is the limit approaching from if x → 3⁺?

Right

Which direction is the limit approaching from if x> or x≥?

Right

A point is only defined if it is _____.

Shaded

What is limit behavior?

The behavior of the value getting close to f(x), but not at it

Constant Multiple Rule

The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function

Difference Rule

The derivative of a difference of functions is the difference of their derivatives

Sum Rule

The derivative of a sum of functions is the sum of their derivatives

Constant Rule

The derivative of any constant function is always 0

What is the limit if lim f(x) = 4 x → -4⁻ lim f(x) = 9 x → -4⁺

The limit does not exist

What do we know about the limit for the following? lim f(x) ≠ lim f(x) x → ⁻ x → ⁺

The limit does not exist (DNE)

What is the following notation asking you to find? lim f(x) x → 2

The limit of f(x) as x approaches 2

Exponent Property of Limits

The limit to and exponent acts the same as constants to an exponent

And empty point, f(x) = 1, f(1), and f(x) = 0 / 0 represent _____ functions.

Undefined

What is ∪?

Unity symbol that joins open intervals

What is the following asking? lim f(x) = x → -4⁻ lim f(x) = x → -4⁺

What is the behavior of the line just barely to the left? Right?

What is the following asking? lim f(x) = x → -4

What is the limit of the function?

When is a limit present?

When both the left and right limits are the same value

A limit is defined by its (x/y) value.

Y

The height "s" at time "t" of a dime dropped from the top of a building is given by s = -16t² + 585. (a). How long will it take the dime to hit the ground? (b). Find the velocity of the dime when it hits the ground.

a. Set s(t) = 0. Solve algebraically. t = 6.05 s b. Find the derivative. Plug in previous answer. -193.6 ft/s

Consider the following. f(x) = (x − 2) / (x² − 4) 1. Describe the interval(s) on which the function is continuous. 2. Identify any discontinuities. 3. If the function has any discontinuities, identify the conditions of continuity that are not satisfied. a. There is a discontinuity at x = c where f(c) is not defined. b. There is a discontinuity at x = c where lim f(x) ≠ f(c). x→c c. There is a discontinuity at x = c where lim f(x) does not exist. x→c d. There are no discontinuities; f(x) is continuous.

a. There is a discontinuity at x = c where f(c) is not defined. b. There is a discontinuity at x = c where lim f(x) ≠ f(c). x→c c. There is a discontinuity at x = c where lim f(x) does not exist. x→c

Y-Intercept

b

Find the derivative of the function. h(x) = x⁹/²

h'(x) = 9x∧(9/2) / 2

What is the notation for finding limits?

lim f(x) x→value

Find the limit. f(x) = x²+1 x≥1 4 x<1 lim f(x) = x → 1⁻ lim f(x) = x →1⁺ lim f(x) = x → 1

lim f(x) = -4 x → 1⁻ lim f(x) = 2 x →1⁺ lim f(x) = DNE x → 1

Slope

m

Find the slope of the graph of the function at the given point. f(t) = t²; (4, 16)

m = 8

How can you determine continuity without a graph?

-Polynomials are always continuous -Find domain restrictions using algebra -Attend to the transitions between pieces in piecewise functions to determine limits

Determine intervals of continuity. h(x) = (x²-1)³

(-∞,0)∪(0,∞)

Determine intervals of continuity. p(x) = 1/(9+x²)

(-∞,0)∪(0,∞)

Find the limit (if it exists). lim (2-x) / (x²-4) x → 2

(2-x) / (x²-4) -x+2 / (x²-4) -x+2 / (x-2)(x+2) -1 / (x+2) -1 / (2+2) -1 / 4

Find the limit. lim (2x+4) / (x²-4) x → -2

(2x+4) / (x²-4) 2(x+2) / (x+2)(x-2) 2 /(x-2) 2 / (-2-2) 2 / -4 -0.05

lim f(x) = (4x² - 8x/x - 2) x → 2

(4x² - 8x/x - 2) 4x(x - 2) / (x - 2) 4x 4(2) 8

Instant Rate of Change

-"IRC" -Rate of change at a specific moment -Derivative

Find the slope of the graph of the function at the given point. f(x) = 2x⁴ − 6x³ + 7x² − 14x; (1, −11)

-10

How do you format open intervals of continuity?

-Only provide x coordinates -(x,y)∪(x,y)

What is the following asking? Find f(-4)

How high is the shaded point?

Differentiation

-The ability to transform a function into its derivative -Function must be continuous and not pointy

Continuity

-The line can be drawn without having to pick up your pencil -Defined in intervals -Continuous if the limits = the value of the function (point)

Tangent Line

-Touches curve of a graph at one point -Slope approximates the rate of change

Find d/dx[7]

0

What is the derivative of a constant? Why?

0; there is no slope, just a horizontal line

What is the derivative of x?

1

Determine where the function has horizontal tangent lines. f(x) = 5x³ - 15x

1, -1

Consider the following. f(x) = x2 − 81, x ≤ 0 9x + 1, x > 0 1. Describe the interval(s) on which the function is continuous. 2. Identify any discontinuities. 3. If the function has a discontinuity, identify the conditions of continuity that are not satisfied. a. There is a discontinuity at x = c where lim f(x) does not exist. x→c b. There is a discontinuity at x = c where f(c) is not defined. c. There is a discontinuity at x = c where lim f(x) ≠ f(c). x→c d. There are no discontinuities; f(x) is continuous.

1. (-∞,0)∪(0,∞) 2. 0 3. a. There is a discontinuity at x = c where lim f(x) does not exist. x→c

Consider the following. f(x) = 16 + x, x ≤ 4 x2 + 4, x > 4 1. Describe the interval(s) on which the function is continuous. (Enter your answer using interval notation.) 2. Identify any discontinuities. 3. If the function has any discontinuities, identify the conditions of continuity that are not satisfied. a. There is a discontinuity at x = c where f(c) is not defined. b. There is a discontinuity at x = c where lim f(x) ≠ f(c). x→c c. There is a discontinuity at x = c where lim f(x) does not exist. x→c d. There are no discontinuities; f(x) is continuous.

1. (-∞,∞) 2. x = DNE 3. d. There are no discontinuities; f(x) is continuous.

Three Ways a Function Can Fail to be Differentibale

1. Discontinuity 2. There is a point 3. Vertical tangent line

The height "s" at time "t" of a dime dropped from the top of a building is given by s = -16t² + 585. Find the instantaneous velocity at t=1.

1. Find derivative 2. Plug in the provided x value 3. Set s(t) = 0 4. Solve algebraically -32 ft/sec

What are the steps to algebraically calculate the derivative?

1. Find f(x + h). Plug into the limit definition of a derivative formula. 2. Find f(x). Plug into the formula. 3. Distribute the "-" in the formula and cancel out the terms. 4. Find the limit.

How do you find the equation of a tangent line?

1. Find the derivative 2. Plug x into derivative to find m 3. Plug x, y, and m into y = mx + b to find b 4. Rewrite the equation without x and y

Find the average rate of change of f(x) over the interval [-3, 1]. Find the instantaneous rate of change when x = -1. f(x) = x² + x - 3

1. Plug each x into the original function to find y 2. Find slope m = -1 1. Find derivative 2. Plug in -1 -1

The height "s" at time "t" of a dime dropped from the top of a building is given by s = -16t² + 585. Find the average velocity on the interval [3, 4].

1. Plug x values into original equation to find y values 2. Plug y values into slope formula -112 ft/s

How do you determine where a function has a tangent line?

1. Set the derivative equal to zero and solve for x 2. Plug x into the ORIGINAL function to find y

How do you find f(x + h)?

1. Take the provided equation, and plug x+h into each X position. 2. Unfoil x² + 2x + 4 (x+h)²* + 2(x+h) + 4 x² + 2xh* + h² - 2x - 2h + 4 *Multiply the coefficient of the exponent value by 2

What are the conditions for continuity?

1. There must be a point 2. The left and right limits must equal each other 3. The point and limit must be in the same spot (all 3 limits are equal)

Find f'(x) f(x) = (4x³ - 2x + 1) / x

8x - x⁻² or 8x 1/x²

Find the marginal cost for producing x units. C = 100(5 + 2√x)

C' = 100x∧⁻½ or 100/√x

A function (can/cannot) be continuous but not differentiable.

Can

A function (can/cannot) be differentiable in an open interval.

Can

A function (can/cannot) be differentiable but not continuous.

Cannot

Calculus is the study of _____.

Change How quickly do things change? How quickly or slowly do things change at different times? How do some quantities change in relationship with other quantities?

How can you move an exponent value from the top or bottom of a fraction?

Change the exponent's sign

What does the following mean? (-∞,0)∪(0,∞)

Continuous everywhere except 0

The slope of the tangent line is the _____.

Derivative of the expression

If the left limit is -2 and the right limit is positive 2, the limit (does/does not) exist.

Does not

Find the limit (if it exists). lim f(s) = s, s ≤ 1 s→1 1 − s, s > 1

Does not exist

What does it mean to differentiate?

Find the derivative

What should you do to find a profit margin?

Find the derivative

Find the limit using direct substitution. lim ³√x+22 x → 5

³√x+22 ³√5+22 ³√27 3