Calculus Review Game- Finding Derivatives

Basic Differentiation

100

The derivative calculates the ________.

Slope or rate of change

200

d/dx 3x²-x+3 =

6x-1

300

Differentiate y=√13x²-5x+8

y′ =26x-5/ 2√13x²-5x+8

400

The cost of producing x toasters each week is given by C = 1785+3x+0.002x² pounds. Find the value of dC/dx when x = 1000, and interpret its meaning.

0.004 (1000) + 3 = 7 pounds. When 1000 toasters are being produced each week, the cost increases by another 7 pounds per toaster.

500

Using quotient rule, differentiate y= (1 + lnx) / (x² - lnx)

y′= [(1/x) - x - 2xlnx] / (x²-lnx)²

Slightly more advanced rules

100

Suppose y = —2x²(x + 4). For what values of x does dy/dx = 10?

x = -1 or -5/3

200

Find the value of z for which the tangent to f(x) = ax⋅ √(1 - x) has gradient a.

x = 0

See HL2 17C #7b

300

y = x³, show that (dy/dx)⋅(dx/dy) = 1

See HL2 17B.2 #8

400

The gradient function of f(x) = (2x - b)^a is
f'(x) = 24x² — 24x +6. Find the constants a and b.

a = 3, b = 1

500

The tangent to f(x) = 2^x + 4^x at point P has gradient ln2. Find the coordinates of P.

(-1, 3/4)

See HL2 17E.2 #4

Properties of Curves

100

State the relationship between speed and velocity.

Speed is the absolute value of velocity, i.e. velocity without direction.

200

Find and classify the stationary points of f(x) = x² / (x + 3).

Local maximum (-6, -12)

Local minimum (0, 0)

Note: x = -3 is a critical point (f' undefined), but not a stationary (f' = 0) nor inflection point (f'' = 0 and change signs).

300

Find intervals where f(x) = -x³ - 6x² +36x -17 is increasing.

-6 ≤ x ≤ 2

400

Find the equation of the normal to f(x) = x + lnx where x = 1.

x + 2y = 3

500

For the function f(x) = ln(x² + 5), find the 

a. turning points

b. points of inflection

c. intervals where it's concave up or down

a. (0, ln5) minimum

b. (-√5, ln10) and (√5, ln10) are non-stationary inflection points

c. Concave up for -√5 ≤ x ≤ √5, down everywhere else.