Math 261 Calculus 1 Test Ch2
Total : 44pts.
Problem 1. (5 pts.) Find the derivative of the function 𝑦 = 𝑥 2 − 3 using defintion of limit.
Problem 2. (5 pts.) Find the equation of the tangent line to the curve 𝑥 2 + 4𝑥𝑦 + 𝑦 2 = 13 at the point (0,1)
Problem 3. (5 pts.) Find 𝑦′′ if 2𝑥 3 − 3𝑦 2 = 8
4𝑥−3 2
Problem 4. (5 pts.) Find the first derivative of 𝑦 = (𝑥 2+1)
Problem 5. (5 pts.) Find the first derivative of 𝑦 = √𝑐𝑜𝑠√𝑥 2 + 4
𝑠𝑒𝑐𝑥
Problem 6. (5 pts.) Find the limit lim 1−𝑠𝑖𝑛𝑥
𝑥→0
Problem 7. (5 pts.) The volume of a cube is increasing at a rate of 10 cm3/min. How fast is the surface area
increasing when the length of an edge is 5 cm?
Problem 8. (9 pts.) The particle moves on a vertical line so that its coordinate at time t is 𝑠 = 𝑓(𝑡) = 𝑡 3 −
12𝑡 + 3 , 𝑡 ≥ 0
Where s is measured in cm and t is measured in second
a) Find the velocity after 2 seconds and acceleration functions after 4 seconds.(2pts.)
b) When is the paticle at rest? (1pt.)
c) When is the particle moving upward and when is it moving downward?(2pts.)
d) Find the total distance that the particle traveled for the first 3 seconds.(2pts.)
e) Draw a diagram to represent the motion of the particle.(2pts.)
Extra Credit (2pts.)
𝑐𝑠𝑐𝑡−𝑐𝑠𝑐𝑥
If 𝑔(𝑥) = lim 𝑡−𝑥 , find the value of 𝑔′(𝜋⁄4)
𝑡→𝑥