Calculus Answers

Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
f(x) =

x2 if −1 ≤ x ≤ 0
7 − 9x if 0 < x ≤ 1

Find the measures of two angles, one positive and one negative, that are coterminal with pi divided by two.

three pi divided by two; negative pi divided by two
pi divided by two + 360°; pi divided by two - 360°
five pi divided by two; negative pi divided by two
five pi divided by two; negative three pi divided by two

Equation for a distance, s(m), travelled in time t(s) by an object starting with an initial velocity u(ms^-1) and uniform acceleration a(ms^-2) is:

s=ut+1/2at^2

a) plot a graph of distance (s) vs time (t) for the first 10s of motion if u = 10ms^-1 and a = 5ms^-2?

b) determine the gradient of the graph at t=2s and t=6s?

c) differentiate the equation to find the functions for
I) velocity (v=ds/dt)
Ii) acceleration (a= dv/dt = d^2 s/dt^2

d) use result from part c to calculate the velocity at t=2s and t=6s?

e) compare your results for part b and d?

Let f be one of the following functions on R. Compute f-hat. (a) f(x)=cosx, xER. (b) f(x)=sinx, xER. (c) f(x)=e^(iax), xER, where aER.

Let f(x) = x^a, xER^n, where a is a multi-index. Compute f-hat.

Let a be a multi-index. Compute the Fourier transform of D^(a)δ.

The legs of a right triangle have lengths A and B satisfying A + B = 10. Which values of A and B maximize the area of the triangle?

The gain of an amplifier is found to be G=20log(10Vout):

The gain of an amplifier is found to be G=20log(10Vout):

Find equations for:
a) dG/dVout ?
b) d^2 G/dV out^2 ?

The same capacitor circuit is now charged up to 12V and the instantaneous voltage is v=12(1- e^-t/2)

The same capacitor circuit is now charged up to 12V and the instantaneous voltage is v=12(1- e^-t/2)

a) Differentiate v with respect to t to give an equation for dv/dt?

b) Calculate the value of dv/dt at t=2s and t=4s?

c) Find the second derivative (d^2 v/dt^2)?

The equation for the instantaneous voltage across a discharging capacitor is given by v=Voe^-t/t , where Vo is the initial voltage and t is the time constant of the circuit.

The equation for the instantaneous voltage across a discharging capacitor is given by v=Voe^-t/t , where Vo is the initial voltage and t is the time constant of the circuit.

a) Draw a graph of voltage against time for Vo =12v and t=2s, between t=0s and t=10s?

b) Calculate the gradient at t=2s and t=4s?

c) Differentiate v=12e^-t/2 and calculate the value of dv/dt at t=2s and t=4s?

d) Compare answers for part b and part c?

e) Calculate the second derivative of the instantaneous voltage (d^2 v/dt^2)?