Calculus Answers

Calculate the area under the curve 𝑦 = 𝑥³ + 4𝑥 + 1 from x=-3 to x=3

1.    Evaluate :        (𝑥³ + 1

𝑥3

) dx.

Evaluate maximum and minimum value of the function

ƒ(𝑥) =  𝑥³-3𝑥² + 3𝑥+1

1.    Also find the equation

of tangent and normal of the ellipse

𝑥2

4

𝑦2

+

16

=1 at the point (-1,3).

b) Evaluate the 2^nd order partial derivatives 62 u

6𝑥2

62 u

and 6𝑦2 if

𝑢 = 2𝑥³+3𝑥² 𝑦 + 𝑥𝑦³+𝑦³.

1.    Find the derivatives of the following functions with respect to x.

𝑥³ + 𝑦³ = 3   

𝑦 = (sin 𝑥)^{𝑡𝑎𝑛𝑥}

A spring is such that a 2 lb weight stretches it by 6 in. An impressed force of F(t) = ¼ sin 8t is

acting on the spring. If the 2-lb weight is released from a point 3 in below the equilibrium point,

describe the motion.

A spring is such that it would be stretched 15.36 in by a 40 lb weight. Let the weight be attached

to a spring and pulled down 6.5 in below the equilibrium point. If the weight is started with an

upward velocity of 7 flt/sec, describe the motion. No damping force but an impressed force of

F(t) = 10lb is present.

A spring with constant 1.5lb/ft, lies on a long smooth (frictionless) table. An 8 lb weight is

attached to the spring and is at rest at equilibrium position. A 6 lb force is applied to the support

along the line of action of the spring for 5 secs and is removed. Discuss the motion.