Calculus Answers

the intensity (I) of illumination given by a projector varies inversely as the square of the distance(d) of its lamp from the screen if the projector is 20m from the screen when the intensity is 2.5 find the distance when the intensity is 62.5 a if r varies inversity as the aquare of h and r =6 when h = 4 

let

f(x) =1 + 2𝑥, 𝑥 ≤ 0

=3𝑥 − 2,0 < 𝑥 ≤ 1

= 2𝑥 ଶ − 1, 𝑥 > 1

i) Check whether f is discontinuous. If yes, find where? ii) Give a rough sketch of the graph of f. 

Find the second derivative of 3y^4+x^7=5x

Determine the length of the parametric curve given by the following parametric equations. x=3sin(t) y=3cos(t) 0<t<2π

Evaluate the following line integrals along with the curve C.

(a) ∫

C (x² − 2y² ) ds; C is the line segment parametrized by r(t) = < t / √ 2 , t / √ 2 >, for 0 ≤ t ≤ 4.

Evaluate the following line integrals along with the curve C.

(a) ∫

C (x² − 2y² ) ds; C is the line segment parametrized by r(t) = < t √ 2 , t √ 2> for 0 ≤ t ≤ 4.

Determine whether the following vector fields are conservative on R 2 . If the vector field is conservative then find a potential function. F =<e^x cos y, −e^x sin y>.

Trace the curve y²=(x-a)(x-b)(x-c) with a, b, and c are all positive.  

a²x²=y³(2a-y) trace the curve

following statement is true and which are false? Give reasons for your answers, in the form of a short proof or a counter example.

Every continuous function is differentiable