Calculus 1 Review

Derivative Rules

100

What is the product rule?

f'g+g'f

200

What is the quotient rule?

f'g-g'f/ g²

300

What is the derivative of a^x?

(a^x ) ln(a)

400

What is the derivative of log_ax?

1/(xlna)

500

Write the chain rule

f'(g(x))g'(x)

Trig Derivatives

100

Derivative of sinx

cosx

200

Derivative of tanx

sec²x or 1/cos²x

300

Derivative of cscx

-cscxcotx or -cosx/sin²x

400

Derivative of tan^-1x

1/(1+ x²)

500

Derivative of csc^-1x

1/ (lxl ((x²-1)^1/2)

Components of the Derivative

100

What does the graph of the derivative describe?

It shows all the slopes of lines tangent to the original graph

200

What does it mean when f'(x)= 0

A maximum, minimum, or critical point

300

How do you decided whether a critical point is a maximum or minimum?

Positive to negative = max; negative to positive =min

400

What does it mean when f''(x)= 0

It is a candidate for a point of inflection.

500

How do you figure out when the original graph, f(x), is concave up?

When f''(x)>0

Things that make you go HHMMM...

100

For f(x)= 2x³-3x²-12x+18, the graph decreasing and concave up over the interval (-1, .5). What is true about the first and second derivatives?

The first derivative is negative and the second derivative is positive.

200

If f(x)= x²-4, and g is a differentiable function of x, what is the derivative of f(g(x))

2g'(x)

300

Differentiate: 3^{sin(5x^2)}

(ln3)3^{sin(5x^2)}cos(5x^2)10x

400

lim as x approaches 0 for (1 - cosx)/x²

1/2

500

The Mean Value theorem for derivatives

What is f'(x) = (f(b) - f(a))/(b - a)

Calculus Fun Facts

100

What is the name of that guy who made the rule about limits that are 0/0

l'hopital

200

What does calculus mean in latin?

small stone

300

What is Mr. Hernandez's favorite hobby?

Pickleball

400

Why is the derivative of x equal to 1?

the line y=x, the slope is always one

500

Who is the real father of Calculus?

Newton or Leibniz - the debate will never end.