Unit 2
Find the derivative of y=x2sin(x)
y'=2xsinx+x2cosx
With the Velocity function of a particle moving on the x axis represented with t seconds and v(t) as in meters per second, given as v(t)=2x if at t=0 the position of the particle is 12 meters, what is the position of the particle at t=8.
The position of the particle at t = 8 is 136 meters because that is the value you get after plugging in 8 for t.
Find relative minimum from F(x)=x3-3x on the interval -2≤x≤2
x=1
What is the correct u sub for this equation? (integral)5sin(5x) dx
U=5x
What is the derivative of sin(x)?
cos(x)
Find lim F(x)=sinx/2x
x-)0
1/2
Identify if the function f(x)=6x3+7x2-5x-2 is increasing or decreasing at the point x = 0. Why?
f(x) is decreasing at x=0 because the derivative is negative.
Find the left riemann sum of f(x)=x2 from 0 to 4 using 4 equal rectangles.
14 Units
What is the difference between a function being continuous and differentiable?
The difference between a continuous and differentiable function is the differentiable function is continuous on the original function and the derivative. The continuous function is only continuous on the original function, not the derivative.
Using tangent line: Using the function f(x)=2x3+4x-7, estimate the value of f(5) using the tangent line from the slope at f(3)
f(5) = 175
A car starts a trip at mile marker 0 and ends at mile marker 110. The trip takes a total time of 7,200 seconds. If the speed limit is 60 mph the whole way, can a police officer confidently give the driver a speeding ticket? Why or why not?
No because the average speed on the time interval of the car is 55 mph, meaning the officer has no way to prove that on any way through that time that the car was speeding.
On a graph that is concave down from 0≤x≤5, what is the correct order of left, right, and midpoint riemann sums, as well as the integral from least to greatest.
Left < Integral < Midpoint < Right
Find f'(x) if f(x) = 7x4-3x2+5x-8
f'(x) = 28x3-6x+5
Using the function f(x)=3x4-2x3+x-10, estimate the value of f(8) using the tangent line from the slope at f(4)
f(8)=3326 (as an estimate).
Is the function f(x)=3x5+8x4-2x3-14x2-7x+8 concave up or concave down at x=1? Why?
f''(1)=166 and 166>0, so the graph of f is concave up at x=1.
Evaluate: (Integral from 1 to 4) 3x2+8x-11 dx
90
If y=(x2+3x)/(x-1) find dy/dx
dy/dx = (x2-2x-3)/(x-1)2
A particle moves on the hyperbola xy=15 for time t > 0 seconds. At a certain instant x=3 and dx/dt=6. What is true about y at this point?
Given the relation xy = 15 differentiate both sides with respect to time t: x(dy/dt)+y(dx/dt)=0 at the instant when x=3, 3y=15 so y=5 then you sub x=3, y=5, dx/dt=6 then plug in the numbers to get 3(dy/dt)+5(6)=0 so dy/dt=-10, Y is decreasing at 10 units per second.
Find the absolute minimum of f(x)=x5-11x3+2x2-12x+6 on the interval -2≤x≤2.
y=-66 at f(2)
The rate at which boxes are loaded into a shipping container per hour is defined by f'(x)=4x2-12x+6. Solve and interpret the meaning of (integral from 0 to 8) f'(x) dx
About 347 boxes are loaded into the shipping container 8 hours after starting.