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.
b) Find the absolute maximum and minimum values of the function h(x)=x3−6x2+9xh(x)=x3−6x2+9x on the interval [1,4][1,4].
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