Section 2.3 flashcards

What is the power rule?

x^n = nx^n-1. Multiply by the exponent, then subtract 1. Ex: f(x) = x^5 = f’(x) = 5x^4.

What is the constant multiple rule.

cf(x) = cf’(x). Leave the constant alone and differentiate everything else. Ex: 7x^4 = 28x³

What is the Sum/Difference Rule?

Differentiate each term separately. Ex: 4x³ - 2x + 8 = 12x² - 2

What should you ALWAYS do before taking a derivative?

Rewrite first, derive second. 1. rewrite radicals, rewrite fractions, simplify exponents, then differentiate.

How do you rewrite radicals?

xn/2 = x^n-1/2. Ex: sqrtx = x^1/2, 4xsqrtx = 4x^3/2, 7x²sqrtx = 7x. (If same base, add exponents)

How do I rewrite fractions?

Move the denominator to the numerator using a negative exponent.

What exponent rules should I remember before differentiating?

If my final answer cannot have negative or fractional exponents, what do I do?

Convert back to radicals and fractions.

How do I solve a tangent line problem?

1. Find the derivative, Plug the given x-value into the derivative to find the slope m, 3. Use the given point (x, y), 4. Plug into y = mx+b, 5. Solve for b. Memory: Derivative —> Slope —> Tangent Line.

How do I solve horizontal tangent problems?

1. Fine the derivative, Factor if possible, Set f’(x) = 0, 4. Solve for the x-values. Memory: Horizontal tangent = slope = 0.

What does “marginal” mean?

Marginal = Derivative

What is the Profit Function?

P(q) = R(q) − C(q)

What is the Marginal Profit Function?

MP(q) = P′(q) = R′(q)−C′(q)

Which keywords tell me I should take the derivative?

• Marginal revenue

• Marginal cost

• Marginal profit

• Growth rate

• Instantaneous rate of change

• Velocity

• Acceleration

• Rate of change

• Slope of the tangent line

• Tangent line

• Horizontal tangent

Memory: If it says rate, marginal, velocity, acceleration, tangent line, or horizontal tangent, think: "Derivative first."

How do I find the average rate of change from x = a to x = b?

Use the original function: f(b) - f(a)/b-a

Tip: DO NOT take the derivative for average rate of change. Memory: Avg rate = two x-values = use the original function.

How do I know whether to use the original function or derivative?

Memory: Average = between two points

Instantaneous = at one point.

How do I find a tangent line using the definition of derivative?

Memory: Point from original, Slope from definition. Line from y = mx + b.

What Does h→0 Mean?

It means plug in h = 0 after simplifying and canceling h. Ex:

How do I find the slope of the tangent line at x = a?

What is the difference between “find the slope” and “find the tangent line equation”?

How do I find where the tangent line is horizontal?

Horizontal tangent means slope is 0: f′(x) = 0

Steps:

1. Find f′(x)

2. Set f′(x)=0

3. Factor if possible

4. If it does not factor nicely, use the quadratic formula

Memory: Horizontal tangent = deriviative ewuals o.

What do I do if f′(x)=0 gives a quadratic that does not factor nicely?

Use the quadratic formula:

How do I rewrite radicals as exponents?

Memory: Power on top, root on bottom.

Classwork Decision Chart: How do I know what method to use?

What does (a, f(a)) mean in a tangent line problem?

means the point on the graph: a = the given x-value, f(a) = the y-value from the original function. Example: If: f(x) = 2x² and x = 4, then f(4) = 2(4)² = 32, so: (a, f(a)) = (4, 32).

How do you turn exponents into radicals?

Formula: