Homework #5
Calculus I 3450:221
Dr. Norfolk
1. Find the derivative of y = (3x + 2 sin(5x2 ))7 .
Solution: y 0 = 7(3x + 2 sin(5x2 ))6 · (3 + 20x cos(5x2 ))
2. Find and simplify the derivative of h(t) = (t4 − 1)3 (t3 + 1)4 .
Solution: h0 (t) = 3(t4 − 1)2 · 4t3 (t3 + 1)4 + (t4 − 1)3 · 4(t3 + 1)3 · 3t2
= 12t2 (t4 − 1)2 (t3 + 1)3 (2t4 + t − 1)
3. If the position on a line of a particle at time t is given by s = A cos(ωt + δ), the particle is
said to undergo simple harmonic motion.
(a) Find the velocity at time t.
(b) When is the particle at rest?
Solution:
(a) v(t) = s0 (t) = −Aω sin(ωt + δ)
(b) When v(t) = 0, which is when ωt + δ = πn (n an integer), so t =
πn − δ
.
ω
4. If g(x) + x sin(g(x)) = x2 − 1 and g(1) = 0, find g 0 (1).
Solution: Via implicit differentiation:
g 0 (x) + sin(g(x)) + x cos(g(x)) · g 0 (x) = 2x
Substituting x = 1 gives
g 0 (1) + sin(0) + cos(0) · g 0 (1) = 2
from which g 0 (1) = 1.
5. Use implicit differentiation to find an equation for the tangent line to the curve x2 + 2xy −
y 2 + x = 2 at the point (1,2).
Solution: Differentiating gives 2x + 2y + 2xy 0 − 2yy 0 + 1 = 0.
Substituting x = 1 and y = 2 gives
2 + 4 + 2y 0 − 4y 0 + 1 = 0, so y 0 = 7/2.
7
Hence, the equation is y − 2 = (x − 1).
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