Recall the Derivative!
(5 seconds)
d/dx(x^n)
nx^(n-1)
(30 seconds)
lim_(t->-3)(6+4t)/(t^2+1)
-3/5
(1 minute 30 seconds)
d/dx((sqrt(x)+x^(9/4)-1)/x^2)
-3/2x^(-5/2)+1/4x^(-3/4)+2x^(-3)
(15 seconds) If r is a positive rational number and c is any real number then,
lim_(x->oo)c/x^r=?
0
(15 seconds)
lim_(x->0)sin(x)/x
1
(5 seconds)
d/dx(c)
0
(45 seconds)
lim_(x->oo)(2x^4-x^2+8x)/(-5x^4+7)
-2/5
(1 minute)
d/dx(e^(1-cos(x)))
sin(x)e^(1-cos(x))
(15 seconds) Fill in the following statement with either continuous or differentiable:
If f(x) is (1) x=a then f(x) is (2) at x=a
1) differentiable
2) continuous
(1 minute) A rock's height in meters while falling off a cliff is given by the function h(t). What is the rock's acceleration in meters per second at t=5 seconds? (Hint: The acceleration is the second derivative of the position)
-9.8 m/s^2
(10 seconds)
d/dx(secx)
secxtanx
(30 seconds)
lim_(x->oo)arctan(x)
pi/2
(1 minute 30 seconds) Do not simplify!
d/dx(x/(1+sqrtx))
((1+sqrtx)*1-x*(1/(2sqrtx)))/(1+sqrtx)^2
(20 seconds) A function f(x) is said to be continuous at x=a if:
?=f(a)
lim_(x->a)f(x)
(1 minute 30 seconds) Given that the below function has a y-intercept at (0,25) find the equation of the tangent line to that function at that point.
y=16e^x-sinx+4x+9
y=19x+25
(15 seconds)
d/dx(log_a(x))
1/(xlna)
(1 minute)
lim_(h->0)(sqrt(9+h)-3)/h
1/6
(1 minute) Use implicit differentiation to solve for dy/dx:
e^x-sin(y)=x
dy/dx=(1-e^x)/-cos(y)=(e^x-1)sec(y)
(30 seconds) Suppose that f(x) is continuous on [a,b] M be any number between f(a) and f(b). Then there exists a number c such that,
1. a<(1)<b
2. (2)=M
1) c
2) f(c)
(2 minutes) Find C such that the piecewise function below is continuous.
f(x)=x^2-C if x<=5,
f(x)=(4x-20)/(x^2-6x+5) if x>5
C=24
(15 seconds)
d/dx(cos^-1(x))
-1/sqrt(1-x^2)
(1 minute 30 seconds)
lim_(x->0)x^2cos(1/x)
By using Squeeze theorem, we find the limit is equal to
0
(2 minutes)
d/dx(ln((xsinx+1)^2))
(2(sinx+xcosx))/(xsinx+1)
(30 seconds) Fill in the blank for the limit definition of the derivative:
lim_(h->0)?=f'(x)
(f(x+h)-f(x))/h
(2 minutes) Find the derivative dy/dx for the following function (Hint: Use logarithmic differentiation).
y=x^x
dy/dx=y(1+lnx)=x^x(1+lnx)