Math 131 Exam 2 Review

Know the limit definition of the derivative as a function. Be able to state it for any function. Be able to work it out for .

Know the definition of the tangent line. y = f '(a)(x - a) + f(a), and know where a function is not differentiable.

1. Find the equation of the tangent line to the given function at the given x-value.

2. Find all values of x at which f is not differentiable. Look for discontinuities, vertical tangents, corners, cusps.

a)

b)

c)

3. The tangent line to f(u) at u=2 is y= 3(u - 2) + 5. The tangent line to g(x) at x=1 is

y= - 4(x-1)+2. Find the tangent line to f(g(x)) at x=1.

4. Find each derivative.

b) Use logarithm rules before differentiating!

c)

d)

e)

f)

5. Find anyl local max, min and inflection points of the function with the given derivative. Show all sign charts. If the function is undefined at a point, that point cannot be an inflection pt.

a)

b) f has V.A. x= -2

c) f has V.A. x=3

6. An object travels in a straight line. f(t) is the distance in meters from a reference point at t seconds.

At 2 seconds, the object is moving farther to the right of the reference point and is decelerating. At t=2, the tangent line has a______________slope and the graph is concave ______________.

The object is still moving to the right at t=5 seconds, it speeds up until t=6 seconds and then slows down. At t=6, the graph of f(t) has __________________.

7. Find each derivative, simplify the function first:

a) b)

8. Find each derivative:

a) b) c)

Key

1.

2. a) f(x) is continuous and differentiable at x=3, continuous but not differentiable at

x=5. f(x) has a corner at x=5.

b) At x=8, f is not continuous so also not differentiable.

At x=9, f is continuous but has a corner so not differentiable.

c) The derivative has vertical asymptotes at x=4 and x=-4, f is continuous everywhere

and has vertical tangents at 4 and -4, not differentiable at 4 and -4.

3. y = -12(x-1)+5

4.

5. a) local min at x=1, inflection points at x=-2 and x=-4/5

b) no local max or min, inflection points at x=1 and x=7

c) local min at x=0, inflection point at x= -3, concavity changes at x=3 but there is no point there.

6. tangent line has a positive slope and the graph is concave down.

at t=6 the graph has an inflection point.

7.

8.