Differential and Integral Calculus
1. Use transformations to sketch the graphs of the following functions. Explain each step.
y = ln (x − 2) + 3 y = − ln (−x)
y = | ln (x)| y = ln |x|
2. If f(x) = 4x − 2 and g(x) = 1
x+2 , express the following as rational functions:
fog(x) gof(x) gog(x)
3. Describe the motion of a particle that is at position (x(t), y(t)) at time
t ≥ 1, if x(t) = ln t and y(t) = sin t.
4. Evaluate each of the following limits or explain why it does not exist:
x+1 limx→0 ln(1 + x) sin2
5. For what value(s) of the constant a is f(x) =
ax + 1 if x ≤ 1
2 − 1 if x > 1
continuous on (−∞, ∞)?
6. Use the Intermediate Value Theorem to prove that there is a positive
number c such that c
2 = 2.
7. For each of the following functions:
f(x) = x + 3
2 − 2x − 15
g(x) = x+3
x2−2x−15 if x 6= −3
−1 if x = −3
(a) Sketch its graph.
(b) Find all values of x for which the function is NOT continuous.
(c) Say why the definition of continuity is not satisfied for those values
8. Let f(x) = x
limx→0 f(x) limx→2 f(x)
limx→−5 f(x) limx→∞ f(x).
9. Find the derivative dy
dx for each of the following:
y = cos 3x
2 y = x
y = sec2 3x y = sin (sin (sin x)) y = x
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