Limits
The function f has the property that as x gets closer and closer to 5, the values of f(x) get closer and closer to 9. Which of the following statements must be true?
limx→5f(x)=9
Derive: x4+5x2+52
4x3+10x
Let f be a differentiable function. If h(x)=(1+f(3x))2, h'(x)=
h′(x)=2(1+f(3x))⋅f′(3x)⋅3
Oil is spilled onto a kitchen floor. The area covered by the oil at time t is given by the function A, where A(t) is measured in square centimeters and t is measured in seconds. Which of the following gives the rate at which the area covered by the oil is changing at time t=7 ?
A'(7)
limx→0 (cosx+3ex)/2ex
2
The function f is given by f(x)=1+3cosx. What is the average rate of change of f over the interval [0,π] ?
-6/π
What is the slope of the line tangent to the curve y3−xy2+x3=5 at the point (1,2) ?
1/8
f(x)=eπx
Find the first and second derivative of the function.
f′(x)=πeπx
f′′(x)=π2eπx
Let f and g be functions such that limx→4g(x)=2 and limx→4f(x)g(x)=π. What is limx→4f(x) ?
2π
If f is the function defined by f(x)=4√x, what is f′(x)?
(1/4)x-3/4
If y=2lnx, then d4y/dx4=
−12/x4
A particle moves along the x-axis so that at time t≥0 its position is given by x(t)=2t3+3t2−36t+50. What is the total distance traveled by the particle over the time interval 0≤t≤5 ?
233
Evaluate limy→7(y2−4y−21)/3y2−17y−28), if it exists.
2/5
f(x)=(1/4)x3-3x+21, what does f'(2)=?
0
Find the derivative of sin-1(3/x2).
-6/√1-(9/x4)
The position of a car is given by the following function:
f(x)=cos(x)e11x+csc(x)
What is the velocity function of the car?
−sin(x)e11x+11cos(x)e11x−csc(x)cot(x)
Evaluate limz→4(√z−2)/(z−4), if it exists.
1/4
(d/dx)(cscx)=
-cscxcotx
Let y=f(x) be a twice-differentiable function such that f(1)=4 and dy/dx=4√y2+5x2. What is the value of d2y/dx2 at x=1 ?
37.494
A rectangle has width w inches and height h inches, where the width is twice the height. Both w and h are functions of time t, measured in seconds. If A represents the area of the rectangle, what would give the rate of change of A with respect to t ?
dA/dt=4h(dh)/(dt)in2/sec