Calc test 2 (1)

Logarithmic differentiation

100

What should you do first when using logarithmic differentiation

Take the natural logarithm of both sides

200

y = x^x

y^l = x^x(lnx+1)

300

Differentiate: y = (x²+1)³ / (x-4)²

y^l= (x²+1)³ / (x-4)² (6x / x² + 1  - 2 / x-4)

400

Differentiate: y = x^sinx

y^l = x^sinx(cos(x)lnx + sin(x) / x)

500

Use logarithmic differentiation to find y^l : y = sqr(x²+1) (x-2)⁵  / e^3x(x+1)⁴

y^l = sqr(x²+1) (x-2)⁵  / e^3x(x+1)⁴  (x/x²+1  + 5/x-2  - 3 - 4/x+1)

Inverse functions

100

What is teh relationship between the graphs of f and f^-1

They are reflections across the line y=x

200

State the derivative formula for an inverse function

(f^-1)(a) = 1 / f^l(f^-1(a))

300

If f(2) = 5 and f^l(2) = 4 find (f^-1)^l(5)

Since f(2) = 5 then f^-1(5) = 2, therefore f^-1 (5) = 1/4

400

Find the inverse of: f(x) = 3x-7

Let y=3x-7. Solve for x: x = y+7 / 3 thus, f^-1 (x) = x+7/3

500

If f(x) = x³ + 1 find (f^-1)^l(9)

First solve f(x) = 9 -> x³ + 1 = 9 -> x³ = 8 -> x=2

Then, f^l(x) = 3x+2 so f^l(2) = 12 -> (f^-1)^l(9) = 1/12

Inverse-trig derivatives

100

Find the derivative: f(x) = arcsin(x)

f^l(x) = 1 / sqr(1-x²)

200

Find the derivative: f(x) = arccos(x)

f^l(x) = - 1 / sqr(1-x²)

300

Find the derivative: f(x) = arctan(x)

f^l(x) = 1 / 1+x²

400

Differentiate: f(x) = arctan(3x)

f^l(x) = 3 / 1+9x²

500

Differentiate: f(x) = arcsin(x²)

f^l(x) = 2x / sqr(1-x⁴)

Implicit differentiatioon

100

Differentiate implicitly: x²+y² = 25

2x + 2y(dy/dx) = 0

So, dy/dx = -x/y

200

Differentiate: x³+y³ = 10

3x²+3y²(dy/dx) = 0

Therefore, dy/dx = - x²/y²

300

Find the slope of the curve: x²+y² = 25 at the point (3,4)

Since dy/dx = -x/y

We get dy/dx = -3/4

400

Find dy/dx: xy+y² = 6

Differentiate: x(dy/dx) + y + 2y(dy/dx) = 0

Therefore, dy/dx = -y / x+2y

500

Find the tangent line to: x²+y² = 25 at (3,4)

The slope is m = -3/4

Using point-slope form: y-4 = -3/4 (x-3)

Therefore in point-slope form: y = -3/4x + 25/4

Tangent lines and related rates

100

What is the slope of the tangent line to f(x) at x=a

m = f^l(a)

200

Find the tangent line to f(x) = x² at x=2

The point is (2,4) and the slope is f^l(2) = 4

Thus, y-4 = 4(x-2) or y= 4x-4

300

Find the tangent line to f(x) sqr(4x² + 5) at x=1

The point is (1,3)

The derivative is f^l(x) = 4x / sqr(4x²+5)

So f^l(1) = 4/3

Therefore, y-3 = 4/3 (x-1)

400

A circle's radius increases at 3cm/s. How fast is the circumference changing

C = 2(pi)(r)

We have dC/dt = 2(pi)(r) (dr/dt)

Thus, dC/dt = 6(pi) cm/s

500

A 10fit ladder leans against a wall. Its bottom moves away from the wall at 2ft/s. How fast is the top moving when the bottom is 6ft from the wall.

The ladder relationship is x²+y² = 100

When x=6 then y =8

Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0

Substitute: 2(6)(2) + 2(8)(dy/dt) = 0

dy/dt = -3/2 ft/s