Basic Derivatives
Find the equation of the line tangent to y=x3 at x=-2.
y=12(x+2)-8
A particle's position function is given by s(t) = t3 - 3t2 + 6t. What is the particle's instantaneous velocity at t = 2 seconds?
v(2)=12 units/sec
What is speed?
speed=|velocity|
Given f'(x)=1/x and f(2)=5. Find the equation of the line tangent at x=2.
y=1/2(x-2)+5
A car is moving along a straight road. Its velocity function is given by v(t) = -2t + 4. When is the car at rest?
t=2
Name two other ways to represent the derivative
1. Instantaneous Rate of Change
2. Slope of the tangent line.
Find the equation of the tangent line of f(x)=3sqrt(x)+4 at x=4.
y=3/4(x-4)+10
A rocket is launched from Earth. Its position function is given by s(t) = t^3 + 3t^2 - 12t + 6, where t is measured in seconds and s(t) is measured in meters. What is the rocket's instantaneous acceleration at t = 4 seconds?
t=36
How does velocity relate to position and acceleration?
s(t)=v'(t)
v(t)=a'(t)
Find the equation of the line tangent to f(x)=3cosx +x at x=pi/2.
y=-2(x-pi/2)+pi/2
A particle moves along the x-axis. The function x(t) gives the particle’s position at time t>0.
x(t)=1/4t4-4/3t3-3t2+5t
At t=4, is the particle speeding up, slowing down, or neither? Justify.
slowing down since v(t)<0 and a(t)>0
What are three ways a function is not differentiable?
1. discontinuity
2. vertical tangent line
3. cusp (sharp turn)
Find the derivative of 1/x3+ sqrt(x).
f'(x)=-3/x4+1/(2sqrt(x))
Find the derivative of f(x)=(2x-3)sinx
f'(x)=2sinx+(2x-3)cosx
Find the value(s) of x, where x has a horizontal tangent line if f(x) =x3/3+4x2+12x-13
x=-6, -2
A car is driving down a highway at a constant speed of 60 mph. What is its acceleration?
The velocity and acceleration have the same sign.