Calculus

Limits

100

(Limit as X approaches 3) of (X+2)

5

200

(Limit as X approaches 0) of (1/X)

Does Not Exist 

300

(Limit of X as approaches 5) of (8)

8

400

Find limit x→1 f (x) if f (x) = (4 − x²), 

                                          x ≤1

                                          (3x), 

                                          x >1 

Limit x→1^- f (x)= 3

Limit x→1⁺ f (x)= 3

500

Find the Vertical Asymptotes of  (2x+1)/(x²-2x-8)

x=-2 and 4

Derivative

100

Find the derivative of 2x²

4x

200

Find the instantaneous rate of change of y with respect to x for y=3x³+2x²-4 at x=3

93

300

Find the derivative of 2x³-(pi)²

6x²

400

Find the derivative of 3(4x²-3cot2x)⁵

15(4x²-3cot2x)⁴ (8x-3(-csc²2x)(2))

500

Find the equation of the tangent line to the graph of 2x²+(4/x)-1 at x=2

y-9=7(x-2)

Application of the Derivative

100

solve for x:

f(x) = square root of x

x = 0

200

determine if Rolle’s theorem can be applied to the function below. If so, find all values of c such that f’(x)=0.

f(x) = cosx, (pi/2, and 3pi/2)

x = pi

300

Find the absolute maximum and absolute minimum of the function on the given interval:

f(x) = 2x³+ 3x² -12x on (-3,2)

f(-3) = 9

f(-2) = 20 (max)

f(1) = -7 (min)

f(2) = 4

400

find the intervals on which the function, f(x) = 6x³-3x²-36x+5, is increasing, decreasing, concave up, concave down, and the x-coordinates of all relative extreme and points of inflection. 

X = -2, 3

inc = (-infinity,-2);(3,infinity)

dec = (-2,3)

max at x = -2

min at x = 3 

CU = (1/2, infinity)

CD = (-infinity, 1/2)

POI at x = 1/2

500

At time t=0 years, a lake has a population of 1500 fish. If the rate of growth of the population modeled by  R(t) = 2000e^0.23t fish per year, what is the population at time t=3?

10,141.0046 per year

Integrals

100

Solve the integral from 0 to 1 of (2x + 4x²) dx

2.333

200

Estimate the area under the curve f(x)= 1/x using a left Riemann Sum and 3 sub intervals.

11/6 or 1.8333

300

Find the average velocity of the function with an acceleration a(t)= 3t+1 and initial velocity v(0)=2 on the interval [1,4].

Average velocity = 15

400

The roof of the walls of the new medical building in Penn State is shaped by the curve H(x) = 30-(x⁸/2,500,000). If H(x) indicates the height in feet, what is the average height of the building? 

26.668

500

At time t=0 years, a lake has a population of 1500 fish. If the rate of growth of the population modeled by  R(t) = 2000e^0.23t fish per year, what is the population at time t=3?

10,141.0046 per year