Topic 3 - Term Test 1

Question 1

3.1

Equation of derivative with f(x)

Answer 1

f’(x) = lim as x approaches a of

f(x) - f(a)
_________
x - a

Question 2

3.1

Equation of derivative with f(a+h)

Answer 2

f’(x) = lim as h approaches 0 of

f(a+h) - f(a)
___________
h

Question 3

3.2

f(x) = abs(x-10)

Evaluate limit x approaches 10-
f(x)-f(10)
_________
x-10

Answer 3

Write absolute as piecewise and sub in case where x is less than 10 and then cancel term out with denom and should equal -1

Question 4

3.1

Derivative of abs(x) at x = 0

Answer 4

Use definition of the derivative and you get abs(x)/x

Solve for limits and because they are not equal the derivative does not exist

Question 5

3.2

Two tangent lines to curve of y = x^2 that pass through the point (3/4,0)

Answer 5

1. Sketch the graph and label
2. Point we are looking for is a, a^2 (sub in a into the function which is x^2
3. Slope would be f’(a) which is 2a
4. Slope is also the rise/run between 3/4, 0 and a, a^2
5. Set these two equal and solve for a, which is x value of the point of tangent line
6. Now sub in f’(a) with the value from 5 to get the slope of the tangent line at the derivative
7. Sub in y = mx + b format to find b