Calculus 1 Worksheet - Part 9
43. Limits:
a) Evaluate the following limit:
limx→0tan(3x)xx→0limxtan(3x)
b) Find the limit:
limx→∞(1+5x)4xx→∞lim(1+x5)4x
44. Derivatives:
a) Find the derivative of the function f(x)=cos(x)x2+1f(x)=x2+1cos(x).
b) Determine the critical points of the function g(x)=2x3−9x2+12x−1g(x)=2x3−9x2+12x−1.
45. Integration:
a) Evaluate the definite integral:
∫14(2x+1)2 dx∫14(2x+1)2dx
b) Find the area between the curves y=exy=ex and y=ln(x)y=ln(x) from x=1x=1 to x=2x=2.
46. Related Rates:
A rectangular box has dimensions xx, yy, and zz, where xx, yy, and zz are all changing with time.
If dxdt=2dtdx=2, dydt=3dtdy=3, and dzdt=4dtdz=4, find the rate at which the volume is
changing when x=3x=3, y=4y=4, and z=5z=5.
47. Exponential and Logarithmic Functions:
a) Solve for xx: 2e2x=162e2x=16.
b) Differentiate the function y=ln(3x−1)y=ln(3x−1).
48. Applications of Derivatives:
a) A rectangular garden is to be fenced off along a riverbank, using the river as one side of the
rectangle. If 120 meters of fencing is available, find the dimensions that will maximize the area
of the garden.
