Calculus Answers

What are the local maxima, minima, and saddle points of the following function? What are the values of the function at those points? f(x, y) = (1 +xy)(x+y)

By about how much does the function f(x, y, z) =e^x cos(y-z) change as a point P(x, y, z) moves away from the origin a distance of ds= 0.1 in the direction of i+ 2j+3k? (Hint: Consider how we can get the differential df from the differential ds.)

Find the directional derivative of the function f(x, y, z) = 2xy-yz at the point(1,1,1) in the direction of u=<1,2,3>. Is there a direction (^v) in which f(x, y, z) has a directional derivative D^vf=-3 at the point (1,-1,1)?

If f(x, y) =x^2-xy+y^2, determine ∇f(1,2) and use it to find the equation of the tangent line to the level curve f(x, y) = 7 at the point (1,2).

The equation for a distance s(m), travelled in time t(s) by an object is given by:

s=2t2-3t+5
The tasks are to:

a) Plot a graph of distance (s) vs time (t) for the first 5s of motion
b) Determine the gradient of the graph at t=2s and t=4s.
c) Identify the position of any turning points and whether they are maxima, minima or points of inflexion.
d) Differentiate the equation to find the functions for
i) Velocity (v=dsdt)
ii) Acceleration (a=dvdt=d2sdt2)

e) Use your result from part d to calculate the velocity at t=2s and t=4s.
f) Calculate the turning points of the function using differential calculus and show which are maxima, minima or points of inflexion by using the second derivative.
g) Compare your results for part b and part e.

Given that P=(7,5) and Q= (8,8), find the component form and magnitude of the vector PQ

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. 4, -8, and 2 + 5i

Solve the following question step by step in detail.
∫²₀ xcosx dx

solve the following question step bt step in detail.

∫ x²e³ˣ dx

∫ (10׳ - 2sec²x) dx