Calculus 1 Assignment 6
Alex Cowan
[email protected]
Due Tuesday, November 10th at 5 pm
1. Find all local minima, local maxima, global minima, and global maxima of the following functions:
a) sin(x), x ∈ R
b) sin(x), x ∈ ( π6 , 5π
6 )
c) | sin(x)|, x ∈ [−4, 4]
d) 3x4 + 4x3 , x ∈ [−2, 5)
e) x1 , x ∈ R\{0}
2*. a) Suppose that f is continuous on [a, b] and differentiable on (a, b), and that f (a) ≤ f (b). Show that
either (b, f (b)) is a global maximum of f , or f has a critical point on (a, b). (Hint: use the mean value theorem)
b) Give an example of a function f continuous on (0, 1] and differentiable on (0, 1) which does not attain either
a global minimum or a global maximum.
3. Find where the following functions are positive, negative, increasing, decreasing, concave up, and concave
down. Then sketch them.
a) 5x3 − 3x5
b) x − sin(x), x ∈ [0, 4π]
c) log(x4 + 27)
4. Redraw the following functions and sketch their derivatives next to them.
5. A particle is moving along the curve defined by x3 + y 3 − 6xy + 3 = 0. At the time t = 1, the particle
is at the point (1, 2). At that moment, the x coordinate’s rate of change is 5 (in the appropriate units).
a) Use linear approximation to estimate the particle’s x and y coordinates at t = 1.01. Plug this guess into the
equation of the curve to get a sense of how far off this approximation is.
dy
b) Swap the dx
dt and dy you found in part a), to get an essentially randomly chosen linear approximation (instead
of the best one possible). Use this (bad) approximation to guess x and y coordinates at t = 1.01. Plug this guess
into the equation of the curve and see how far off you are.