Calculus Introduction: Home Assignment
Due Date:
Instructions:
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Show all your work for full credit.
Justify your answers and include units where applicable.
Submit your assignment as a single PDF file.
Part 1: Understanding Limits
1. Evaluate the following limits: a. lim⁡�→2(3�2−4�+1)limx→2(3x2−4x+1)
b. lim⁡�→−1�2−1�+1limx→−1x+1x2−1 c. lim⁡�→0sin⁡(�)�limx→0
xsin(x)
2. Explain why the limit lim⁡�→01�limx→0x1 does not exist.
Part 2: Differentiation
1. Find the derivative of the following functions: a.
�(�)=2�3−5�2+4�−7f(x)=2x3−5x2+4x−7 b. �(�)=�−1�g(x)=x−x1
c. ℎ(�)=��⋅ln⁡(�)h(x)=ex⋅ln(x)
2. A particle moves along a line so that its position at time �t is given by
�(�)=�3−6�2+9�s(t)=t3−6t2+9t. Find the velocity and acceleration of
the particle at �=3t=3.
Part 3: Basic Integration
1. Evaluate the following integrals: a. ∫(3�2−4�+1)��∫(3x2−4x+1)dx b.
∫01(2�+1)��∫01(2x+1)dx c. ∫����∫exdx
2. The velocity �(�)v(t) of a car is given by �(�)=4�v(t)=4t. Using
integration, find the displacement of the car from �=2t=2 to �=5t=5.
Part 4: Application Problems
1. A tank contains 100 gallons of water, which is being drained at a rate of
�(�)=5�r(t)=5t gallons per minute, where �t is the time in minutes. How
much water will be left in the tank after 10 minutes?
2. A ball is thrown upward with an initial velocity of 20 m/s from a height of 50
meters. The height ℎ(�)h(t) of the ball at time �t seconds is given by
ℎ(�)=−5�2+20�+50h(t)=−5t2+20t+50. Determine the time at which the
ball reaches its maximum height and calculate this maximum height.
Submission Guidelines:
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Compile your solutions neatly and submit them in a PDF format.
Ensure your work is clear and legible, with each problem solution starting on a
new page.
Include your name, date, and assignment number at the top of each page.