PHY 1103 Calculus

CLO 01 FUNCTION

Sketch the graph of different functions (14th,15th,16th,17th,18th)

Find domain, range and inverse of different functions and their definitions.

CLO 1 Limit and continuity

Evaluate limits (13th,14th,16th)

Squeezing theorem(13th,14th,16th)
(δ - ε) definition (17th)
Check whether a function is continuous and differentiable or not
(17th,18th,14th)

CLO 2 Differentiation

State and prove Leibniz's theorem (14th,17th,18th)

Leibniz's theorem related problems (14th,17th,18th)
Differential coefficient / find dy/dx

differentiability

Related rate related problems

A camera mounted at a point 3000 ft from the base of a rocket launching pad.
If the rocket is rising vertically at 800 ft/s when it is 4000 ft above the launching pad, how fast
must the camera elevation angle change at the instant in order to keep the camera aimed at the
rocket?(13th,15th) (Howard Anton 10th ed 3.4 pg 206 related rate ex 4)

A baseball diamond is a square whose sides are 90 ft long. Suppose that a player running from
second base to third base has a speed of 30 ft/s at the instant when he is 20 ft from third base.
At what rate is the player’s distance from home plate changing at the instant? (16th)(Howard
Anton 10th/11th ed 3.4 related rate ex 3)

CLO 3 Application of differentiation

i. State and prove Rolle’s theorem.(14th,15th,16th,18th)

ii. State and prove mean value theorem (16th,17th)

Examine the validity of the hypothesis and conclusion of Rolle’s theorem , M.V.T
Test whether the function f(x)=3+2x−x² satisfies the mean value theorem or not, on the interval
(0,1).(18th,17th)

Verify Rolle's theorem for the function f(x)=x²+5x−6 in the interval (−6,1). (18th)
Verify rolles theorem in the interval… (14th)

Absolute maximum and minimum related problems (14th,17th,18th)

CLO 3 Indeterminate forms (L’Hopital Rule)***

Write down the possible indeterminate forms and derive L’Hopital’s rule.(16th)

lim xlnx
x→0 +
lim (tan𝑥²/x)
⁡𝑥→0
lim (1/𝑥−1/sin𝑥)
⁡𝑥→0
Show that
lim (1+sin⁡x)^1/x=e
x→0
(16th

lim (1+1/x)^x=e
x→

(i)
lim (lntan𝑥/ln𝑥)
𝑥→0

(ii)
lim (𝑒^𝑥− 𝑒^−𝑥 + 2sin⁡𝑥−4𝑥/𝑥⁵)
⁡𝑥→0

etc.
https://drive.google.com/file/d/1dk0cO3hYsdvfJ_d0eAeKT3FEENpGlc31/view?usp=drivesdk

https://youtu.be/kfF40MiS7zA?si=g62-boK9XhAGvJ7W

CLO 4 (Gamma and beta funtion)

 1.​ Prove that Γ(m+1) = mΓ(m) =m!
(15th,16th,18th)
2.​ Establish a relation between gamma and beta functions.
(15th,16th,17th,18th)
3.​ Show that Γ(1/2)= √π​(16th,17th)
Γ(5/2)(18th)
Improper integrals (15th,17th,16th)
Test of convergence

https://drive.google.com/file/d/1biwugQUMvUGFS5OBUQJzn91v232GI5qL/view?usp=drivesdk

CLO 4 INTEGRATION
State and prove the fundamental theorem of calculus (14th,15th,18th)

Evaluate integrals, definite , indefinite integrals (14th,15th,16th,17th,18th)

Integration by reduction(17th,18th)

CLO 05 Applications of Integration

Volumes of solids revolution

i) Find the volume of the solid that results when the region above the x-axis and below the
ellipse x²/a²+y²/b²=1; (a>0,b>0) is revolved about x-axis.
(15th)
(ii) Find the volume of the solid that results when the region above the axis and below the ellipse
x²/9+y²/16=1 is revolved about the x-axis(16th)

(iii) Find the volume of the solid generated when the region under the curve y=x² over the
interval [0,2] is rotated about the line y=−1(17th)[Howard Anton 6.2 example 6]

(iv) Find the volume of the solid that is obtained when the region under the curve y=√x over the
interval [1,4] is revolved about the x-axis.(18th)[Howard Anton 6.2 example 2]
[Howard Anton 6.2 example 1](14th)

Plane Areas
Area of a surface of revolution

(I)Show that the area under a curve f(x) and x-axis is ∫f(x)dx from x=a to x=b (16th)​
(ii) Find the area of the region bounded above by y=x+6, bounded below by y=x² and bounded
on the sides by the lines x=0 and x=2(16th)[Howard Anton 6.1 ex 1]
​
(iii) Find the area of the region that is enclosed between the curves y=x² and
y=x+6(16th)[Howard Anton 6.1 ex 2]

(iv) Find the area common to the two parabolas x² =4ay and y²=4ax (17th)
(v) Find the area of the surface that is generated by revolving the portion of the curve
y=x³ between x=0 and x=1 about the x-axis.(17th)[Howard Anton 6.5 ex 1]

(vi) Explain the notion of surface of revolution. Find the area of the surface that is generated by
revolving the portion of the curve y=x² between x=1 and x=2 about the y-axis.(18th)
[6.5 ,Example 2]

[6.1—>ex 1,2
6.2 —>ex 1,2,6
6.5 —> ex 1,2]

CLO 05 Graphs in polar coordinates and application

Areas enclosed by curves in polar coordinates

Find the area bounded by the cardioid r=a(1+cos⁡θ) (15th)

Find the area bounded by the cardioid r=5(1+cosθ)(16th)

Find the area of the region that is inside of the cardioid

r=4+4cosθ and outside of the circle r=6(17th)(10.3 ex9)

Find the area of the region enclosed by the rose r=4cos3θ.(18th)

Arc length
Find the total arc length of the cardioid
r=2(1−cosθ)(17th)

Calculate the arc length of the polar curve

r=e^3θ from θ=0 to θ=2(17th)

State the formula for arc length for polar curves. Find the total arc length of the cardioid
r=1+cosθ. (18th)(10.3 ex 5)

Find the points on the cardioid r=1−cosθ at which there is a horizontal tangent line, a vertical
tangent line, or a singular point.(18th)
(10.3 ex 2)
) What is meant by parametric equations? In a disastrous first flight, an experimental paper
airplane follows the trajectory of the particle whose parametric equations of motion are
x=t−3sint,y=4−3cost,t≥0, but crashes into a wall at time t=10.
(i) At what time was the airplane flying horizontally?
(ii) At what time was it flying vertically?(18th)
[10.1 example 5]