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Derivatives Review
1.
Name: ___________________
Sketch a tangent line at the given point for the graph.
a. x = -3
b. x = 3
c. x = 8
2.
List the points in order of increasing slope of the tangent line.
B
A
D
C
3.
𝐟(𝐚+𝐡)−𝐟(𝐚)
𝒉
𝒉→𝟎
Use the limit definition: 𝐥𝐢𝐦
, to find the indicated derivative.
𝑓′(2) 𝑓𝑜𝑟 𝑓(𝑥) = 𝑥 2 − 3𝑥
4. Use the position function f(t) -16t2 + 20t + 8 to find the instantaneous velocity and
acceleration at time a = 2.
5. Given the position function s(t) = -16t2 + 50t, give the velocity at t = 5 seconds and tell whether
or not the object is rising or falling at this time.
6.
Use the graph given below to sketch the corresponding graph of its derivative
7. Find a value c satisfying the Mean Value Theorem for f(x) = 2x3 + 3x -1 on the interval [0, 2].
8.
Compute the derivative of:
a. f(x) = 2x4 + 5
b. f(x) = √2𝑥 + 6
c. f(x) = (x2 + 4)(x3-2x + 2)
d. f(x) =
2𝑥−3
6𝑥+1
e. f(x) = sin 5x3
f.
f(x) = 5 tan 3x – 4 csc x
g. f(x) = (2x2 – 5)6
h. f(x) = (x2-1)
i.
𝑥 4 +3𝑥 3
𝑥 3 +2
h(x) = ln (3x2 + 5)
9. Find the derivative of the function; be careful, it is NOT the first derivative you are finding.
a. f’’(x)
for
f(x) = xe3x
b. f(4)(t)
for
f(t) = 2𝑡 2 − 8 +
c. f’’’(x)
for
f(x) = x7 + 7x5 – 6
4
𝑡3
10. Find an equation of the tangent line to y = f(x) at x = a.
a. f(x) = 2x2 – 4
a=3
b. f(x) = x cos x
a = π/3
11. Find a function for the given derivative.
a. f’(x) = 2x – 3
b. f’(x) = √𝑥
c. f’(x) = 5x4
d. f’(x) = 5x4 + 1