Topic 3 - Term Test 2

Question 1

Let f(x) = ae^(x-1) + x^2 + b

Find a and b so the tangent line at (1, 2) has slope 5

Answer 1

Need f(1) = 2 and f’(1) = 5

Find f(1) which should equal a + b = 1

Then find f’(1) and you should get a = 3
So then b = -2

Question 2

Derivatives of the following functions

1. Sinx
2. Cosx
3. Tanx

Answer 2

1. cosx
2. -sinx
3. sec^2x

Question 3

Derivatives of the following functions
4. Cscx
5. Secx
6. Cotx

Answer 3

4. -cscxcotx
5. secxtanx
6. -csc^2x

Question 4

Derivative of a^x

Answer 4

a^x * ln(a)

Question 5

Derivative of logax

Answer 5

1/x*ln(a)

Question 6

Derivative of x^X

Answer 6

Use logarithmic differentiation

ln(y) = ln(x^x)
ln(y) = x(ln(x)
1/y(dy/dx) = ln(x) + 1/x*x
1/y(dy/dx) = ln(x) + 1
dy/dx = (ln(x)+1)(x^x)

Question 7

Derivative logx

Answer 7

1/xln(10)