Derivative of a Function (3.1)
Differentiability (3.2)
Rules for Differentiation (3.3
Velocity and Other Rates of Change (3.4)
Derivatives of Trig Functions
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)
100
Name the four cases where a derivative will not exist. (Think graphically.)
1. corner 2. cusp 3. vertical tangent 4. discontinuity
100
Find the derivative of: f(x)=x^3+4x^2-7x
3x^2+8x-7
100
How does acceleration relate to position?
Acceleration is the second derivative of position
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
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))
200
Define what we mean when we say that a function is locally linear at x=a.
The function resembles its own tangent line very close to 'a'. In other words, we can zoom in on the graph at x=a and it will eventually look linear.
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.
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.
200
Which of the standard six trig functions have negative derivatives?
All that start with C! cos(x), csc(x), cot(x)
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
300
The numerical derivative computed by your calculator uses the symmetric difference quotient. What is the symmetric difference quotient?
(f(a+h)-f(a-h))/2h
300
For what value of x does the function f(x)=x^2-10x have a slope of 20.
x=15
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)
300
Find y' if y=2sin(x)-tan(x)
2cos(x)-sec^2(x)
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
400
True or False. If false, provide a counterexample: If 'f' has a derivative at x=a, then f is continuous at x=a. Is the converse true or false. If false, provide a counterexample.
True! Converse: If 'f' is continuous at x=a, it has a derivative at x=a. False: f(x) = |x|
400
Use the product rule to find dy/dx of: y=4x(cos(x))
-4xsin(x)+4cos(x)
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
400
y=3x+xtanx. Find y'.
3+x sec^2(x)+tan(x)
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.
500
What is the Intermediate Value Theorem for Derivatives?
If a and b are any two points in an interval on which 'f' is differentiable, then the derivative, f', takes on every value between f'(a) and f'(b).
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
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
500
y=(cos(x))/(1+sin(x)). Find dy/dx.
-1/(1+sin(x))
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