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Calculus - Differentiation - drawing derivative curves.
Test Yourself 1.


 

Questions on this page require drawing derivative graphs:
1. Given equations of any of the main types of functions.
2. Given graphs of functions.

 

From given equations.

Draw the graph for each of the given functions in Q 1 - 21.
Then draw the graph of the derivative function either on the same set of axes or underneath depending on which presentation you consider shows your graph clearer.
If you choose to draw separate axes underneath, ensure the y axes are aligned.

Remember, you can use Desmos to help you when you know the function. If not sure how, check the Calculus list of Desmos videos.
Given a parabola. 1. y = (x + 3)(x - 5)
  2. y = (1 - x)(x - 7)
  3. (i) Differentiate to find the gradient at x = 2.

(ii) Hence draw the derivative graph and comment on the placement and the shape of the derivative graph.

(ii) What comment does the value of the derivative curve at x = 4 make about the value of the original curve at that value of x?
Given a cubic. 4. y = x(x - 2)(x + 4)

(i) Draw a sketch of the original curve.

(ii) Draw a sketch of the derivative curve.

(iii) What interpretation is possible for the ordinate values of the original and of the derivative curves at the x value of about -0.7?
  5. y = (x + 2)(1 - 2x)(x - 4)

(i) Draw a sketch of the original curve.

(ii) Draw a sketch of the derivative curve.

(iii) After checking your sketches with those in the solutions, comment on the ordinate values associated with the abscissa values (x values) for x = -0.907, x = 2.573 and x = 0.0833.
  6. y = x2(x - 3)

(i) Draw a sketch of the original curve.

(ii) Draw a sketch of the derivative curve.

(iii) Comment on the features of the original curve in relation to the derivative curve when
x = 0, x = 1 and x = 2.

Given a line

7. Draw sketches of the line y = 3x - 6 and of its derivative.

Comment on the two graphs.
  8. Sketch both y = -2x + 1 and 2y = x + 4
and then sketch the derivative graphs for each line.

What do you notice about the two given lines?
 

9. y = -5.

(i) Draw the curve for both the original function and the derivative function.
(ii) Explain the shape and position of the derivative graph.
Given a hyperbola. 10.
  11.
 

12.

(i) Why is the derivative curve under the x axis for all values of x while the original curve is mostly above the x-axis?

(ii) How does the + 3 term affect the graph of the derivative curve?
Given an absolute value.

13. f(x) = |2x|.

What is the gradient at x = 0?
 

14. f(x) = |x - 3|

What is the gradient at x = 3?
  15. y = 1 - |x + 2|
Given a trig function. 16. y = sin x for x: [-180°, 180°].
  17. y = 2sin x + 1 for x: [-180°, 180°].
  18. (i) Draw y = cos x for x: [0°, 360°].

(ii) Using your graph, draw the graph of y = sec x.

(iii) Hence draw the derivative graph of y = sec2 x on separate axes underneath the first set.

NOTE: This question is not requiring the derivative itself. But it is setting the strategy for work elsewhere of determining the derivative of y = tan x.
Mixed functions (piecewise). 19. y = -2 - x with x: (- ∞, -2]

y = 4 - x2 with x: (-2, 2)

y = x - 2 with x: [2, ∞)

  20. y = x + 1 for x: (-∞ , 0]

for x: (0, 2]
  21.

From given graphs.
Given a graph 22. The graph of y = f(x) is shown below.

(i) Copy the graph onto your page.

(ii) On a set of axes below your graph, draw the graph of the derivative function f '(x).
  23.

(i) Copy the graph shown above.

(ii) Sketch the curve of the derivative function of the curve shown.
  24.

The diagram above shows a graph of y = f(x). In particular, the diagram shows that the function has:

  • a stationary point at x = 1;
  • a point of inflection at x = 3; and
  • a horizontal asymptote at y = -2.

Sketch the graph of the derivative function y = f '(x) to show the implications of the main characteristics of f(x).
  25.
Drawing a second derivative curve. 26. The following diagram shows the graph of a derivative function f '(x) for the domain [-2.5, 4]. Assume f (0) = 0.


(i) Copy this diagram.
(ii) On a separate set of axes (exactly underneath), draw the graph of f ''(x).
  27. (i) Draw an approximate sketch of the function

y = x2 - 4x + 5.

(ii) Draw curves for each of the first three derivatives of the function.
  28. Given the function y = x(x - 2)(3 - x), draw curves for the first three derivatives of the function.