MCV 4U Exponential Logarithmic & Trigonometric Functions Derivatives Worksheet

MCV 4UName ______________________________
ASSIGNMENTT DERIVATIVES EXPO TRIG____/33
Part 1 – Multiple Choice [1 mark each]
_____ 1. 𝑙𝑜𝑔𝑒 𝑝 is the same as
a) 𝑙𝑜𝑔𝑒 𝑝
b) 𝑝𝑒
c) 𝑒 𝑝
d) 𝑙𝑛𝑝
e) none of these
_____ 2. If 𝑓(𝑥) = 𝑠𝑖𝑛3 ⁡(6𝑡), then the derivative of f(x) is
a) 18𝑠𝑖𝑛(6t) cos(6t)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡b) 3𝑠𝑖𝑛2 (6t)
c) 18𝑠𝑖𝑛2 (6t)cos⁡(6t)
d) −𝑠𝑖𝑛2 (6t)t
e) none of these
_____ 3. If 𝑓(𝑥) = 𝑠𝑖𝑛⁡(𝑘𝑥), where k is a constant, then the derivative of f(x) is
a) 𝑘cos⁡(𝑘𝑥)
b) −𝑘cos⁡(𝑘𝑥)
c) −𝑘cos⁡ 𝑥
d) 𝑘cos⁡ 𝑥
e) none of these
_____ 4. The derivative of 𝑓(𝑥) = −3cos⁡(6𝑥) is
a) −18cos⁡(6𝑥)
b) −9sin⁡(6𝑥)
c) 18sin⁡(6𝑥)
d) −3sin⁡(2𝑥)
e) none of these
_____ 5. The derivative of 𝑓(𝑥) = 23𝑥 is
a) 23𝑥 𝑙𝑛2(3)
b) 63𝑥 𝑙𝑛2
c) 23𝑥 𝑙𝑛2
d) 23𝑥−1 (3)
e) none of these
2
_____ 6. The derivative of 𝑓(𝑥) = 66𝑡−2𝑡 is
a) 66𝑡−2𝑡
2
2
2
b) 66𝑡−2𝑡 (6-4t)
2
c) 66𝑡−2𝑡 (𝑙𝑛6)
d) 66𝑡−2𝑡 (𝑙𝑛6)(6 − 4𝑡)
e) none of these
_____ 7. The derivative of f ( x) = 7 x is
a) 7 x ( ln 7 )
b) ln(7) x
d) 7 ln(7) x
c) x ln(7)
e) none of these
_____ 8. The derivative of f ( x) = x4  5x is
c) 5x ( 4 x + ln 5)
b) x3 ( 4 x + ln 5)
a) x 4  5x ( 4 x + ln 5)
d) x3  5x ( 4 x + ln 5)
e) none of these
_____ 9. The derivative of f ( x) = x2  sin x is
a) 2x sin x + 2x cos x
b) 2 x sin x + x 2 cos x
c) 2sin x + x cos x
d) 2sin x + 2x cos x
e) none of these
_____ 10. The derivative of f ( x) = 4e x − 3sinx + 5 is
b) 4e x − 3cos x
a) 4e x − 3cos x + 1
c) e x + 3cos x
d) 4e x + 3cos x
_____ 11. The derivative of 𝑓(𝑥) = 𝑙𝑛2𝑥 2 is
a) 2𝑥𝑙𝑛𝑥
2𝑥
b) 𝑙𝑛𝑥
2
c) 𝑙𝑛𝑥
1
d)⁡𝑥
d) none of these
e) none of these
_____ 12. Determine the slope of the tangent to the curve y = 3(2x) at x= 3
a) 24𝑙𝑛2
b) 8𝑙𝑛2⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡c) −216𝑙𝑛2
d) none of these
_____ 13. The derivative of 𝑓(𝑥) = cos⁡(4𝑥) is
a) −4cos⁡(4𝑥)
b) 4sin⁡(4𝑥)
c) −4sin⁡ 𝑥
d) −4sin⁡(4𝑥)
e) none of these
_____ 14. The derivative of 𝑓(𝑥) = 4sin⁡(𝑥 2 ) is
a) 4cos⁡(𝑥 2 )(2)
b) 8𝑥𝑐𝑜𝑠(𝑥 2 )
c) −2𝑥 2 (𝑐𝑜𝑠𝑥)
d) none of these
_____ 15. The derivative of 𝑓(𝑥) = 7𝑒 4𝑥 is
a) 28𝑒 4𝑥
b) 28𝑒 4𝑥−1
c) 7𝑙𝑛𝑒(4)(𝑒 4𝑥 )
d) 7𝑒 4𝑥
e) none of these
Transfer the answers from the multiple choice questions in this table
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Full Solution. For the following questions show all the steps for full marks
1. Identify the local extrema for 𝑓(𝑥) = 𝑥 2 ∙ 𝑒 𝑥
[4 marks]
2. Determine the slope of the tangent line to the graph of 𝑦 = 3𝑥 2 sin(2𝑥) at x = 3π /4 (exact numbers only) [3marks]
3. The position function of a particle that moves in a straight line is s(t ) = 2 t + cos(2 t ) [5 marks]
a) Find the velocity of the particle at time t
b) For what values of t in the interval 0  t  3 is the particle at rest?
c) What is the maximum velocity of the particle?
4. The value of an investment is modelled by 𝐴(𝑡) = 𝐴0 𝑒 0.065𝑡 , where A is the amount the investment is worth after t
years and A0 is the initial amount invested [6 marks]
a) If the initial amount was $3000, what is the value of the investment after 5 years?
b) How long will take for the investment to double in value?
c) At what rate is the investment growing at the time when its value has doubled?