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The midterm exam for math 151, held on november 2, 2005. The exam covers topics such as derivatives, implicit differentiation, and finding equations of tangent lines. Students are required to find derivatives of given functions, identify graphs of functions and their derivatives, and solve problems related to curves and angles between moving objects.
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(1a) (4 marks) y ′^ and y ′′^ where y = cos(e^2 x)
Answer
Answer
0
20
40
60
y
–3 –2 –1 1 2 3 4 x
0
20
40
60
y
–3 –2 –1 1 2 3 4 x
a b
0
20
40
60
y
–3 –2 –1 1 2 3 4 x
0
20
40
60
y
–3 –2 –1 1 2 3 4 x
c d
(3a) (1 mark) Show that (0, 0) lies on the curve.
Answer
(3b) (2 marks) Use implicit differentiation to find y ′.
Answer
(3c) (2 marks) Find the equation of the tangent line at a point (a, b) on the curve (your answer should contain a and b).
Answer
θ
Answer
f (x) =
x +
x
Answer