Derivatives (Chapter 3)

Derivative of a Function ()

100

Give the full definition of the derivative of a function f with respect to x.

lim h->0 ((f(x+h)-f(x))/h)

200

What is the alternate definition of the derivative of a function f at the point x=a?

lim x->a ((f(x)-f(a))/(x-a))

300

Give five ways to denote the derivative of a function y=f(x).

f'(x) y' d/dx (f(x)) dy/dx df/dx

400

Consider the function y=x^2. Describe its derivative, relating the derivative to the original function.

The derivative is 0 at x=0 (there is a horizontal tangent line). When x<0, f(x) is decreasing and the derivative is negative. When x>0, f(x) is increasing and the derivative is positive. The derivative: y'=2x

500

Without a calculator, how do you know if a function is differentiable at a point?

It should first be continuous at that point. Then, the left hand derivative should match the right hand derivative.

Rules for Differentiation

100

Find the derivative of: f(x)=x^3+4x^2-7x

3x^2+8x-7

200

Why is the derivative of a constant always 0?

If the function is a constant, it is simply a horizontal line. Any line drawn tangent to the function will also be a horizontal line and horizontal lines have a slope of 0.

300

For what value of x does the function f(x)=x^2-10x have a slope of 20.

x=15

400

Use the product rule to find dy/dx of: y=4x(cos(x))

-4xsin(x)+4cos(x)

500

Let y=u*v where u and v are functions. Find y'(3) if: u(3)=3 u'(3)=-4 v(u)= 1 v'(u)=2

y'(3)=2

Velocity and Other Rates of Change

100

How does acceleration relate to position?

Acceleration is the second derivative of position

200

If given the position function, how would you find the speed?

Speed is the absolute value of velocity and velocity is the derivative of position. Therefore, take the absolute value of the derivative of the position function.

300

What should you do if asked for the average rate of change of a function over the interval [a,b]?

Average rate of change is asking for the slope of the secant line which is the usual slope formula. Therefore, you should compute (f(a)-f(b))/(a-b)

400

A particle moves along a line so that its position is given by the function s(x)=t^2-4t+3, where s is measured in meters and t is measured in seconds. Using the correct units, find the displacement of the particle during the first 2 seconds. Give an explanation of your answer. Then find the average velocity of the particle during the first 4 seconds. Use correct units.

Displacement: -4, the object is four units to the left of where it started Average velocity: 0 m/s

500

A particle moves along a line so that its position is given by the function s(x)=t^2-4t+3, where s is measured in meters and t is measured in seconds. Find the velocity and acceleration of the particle when t=4. Use correct units.

Velocity: 4 m/s Acceleration: 2 m/s^2

Derivatives of Trig Functions

100

What is the derivative of sin(x), cos(x), and tan(x)?

d/dx sin(x)=cos(x) d/dx cos(x)=-sin(x) d/dx tan(x)= sec^2(x)

200

Which of the standard six trig functions have negative derivatives?

All that start with C! cos(x), csc(x), cot(x)

300

Find y' if y=2sin(x)-tan(x)

2cos(x)-sec^2(x)

400

y=3x+xtanx. Find y'.

3+x sec^2(x)+tan(x)

500

y=(cos(x))/(1+sin(x)). Find dy/dx.

-1/(1+sin(x))