A function, \(f\) , is given by

\[f(x) = 4 \times {2^{ - x}} + 1.5x - 5.\]

Calculate \(f(0)\)

Use your graphic display calculator to solve \(f(x) = 0.\)

Sketch the graph of \(y = f(x)\) for \( - 2 \leqslant x \leqslant 6\) and \( - 4 \leqslant y \leqslant 10\) , showing the \(x\) and \(y\) intercepts. Use a scale of \(2\,{\text{cm}}\) to represent \(2\) units on both the horizontal axis, \(x\) , and the vertical axis, \(y\) .

The function \(f\) is the derivative of a function \(g\) . It is known that \(g(1) = 3.\)

i)     Calculate \(g'(1).\)

ii)    Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 1.\) Give your answer in the form \(y = mx + c.\)

\(4 \times {2^{ - 0}} + 1.5 \times 0 - 5\)       (M1)

Note: Award (M1) for substitution of \(0\) into the expression for \(f(x)\) .

\( =  - 1\)        (A1)(G2)

\( - 0.538\,\,\,( - 0.537670...)\) and \(3\)        (A1)(A1)

Note: Award at most (A0)(A1)(ft) if answer is given as pairs of coordinates.

(A1)(A1)(A1)(ft)(A1)(ft)

Note: Award (A1) for labels and some indication of scale in the correct given window.

Award (A1) for smooth curve with correct general shape with \(f( - 2) > f(6)\) and minimum to the right of the \(y\)-axis.

Award (A1)(ft) for correct \(y\)-intercept (consistent with their part (a)).

Award (A1)(ft) for approximately correct \(x\)-intercepts (consistent with their part (b), one zero between \( - 1\) and \(0\), the other between \(2.5\) and \(3.5\)).

i)     \(g'(1) = f(1) = 4 \times {2^{ - 1}} + 1.5 - 5\)       (M1)

Note: Award (M1) for substitution of \(1\) into \(f(x)\).

\( =  - 1.5\)        (A1)(G2)

ii)    \(3 =  - 1.5 \times 1 + c\) OR \((y - 3) =  - 1.5\,(x - 1)\)       (M1)

Note: Award (M1) for correct substitution of gradient and the point \((1,\,\,3)\) into the equation of a line. Follow through from (d)(i).

\(y =  - 1.5x + 4.5\)        (A1)(ft)(G2)

Question 6: Functions and Calculus
Many candidates were able to calculate \(f(0)\)  although some had a problem with \({2^0}\). Very few were able to find both roots in part (b), even when they represented correctly both zeros in their sketch. Many sketches were reasonably well drawn, presenting a smooth curve with correct points of intersection. Many lost a mark for not labelling their axes and the given scale was ignored by some, which was not penalized but often resulted in sketches which were difficult to read. The stronger candidates calculated \(g'(1)\) correctly, but very few represented the gradient of the function at \(x = 1\) correctly.

Question 6: Functions and Calculus

Many candidates were able to calculate \(f(0)\)  although some had a problem with \({2^0}\). Very few were able to find both roots in part (b), even when they represented correctly both zeros in their sketch. Many sketches were reasonably well drawn, presenting a smooth curve with correct points of intersection. Many lost a mark for not labelling their axes and the given scale was ignored by some, which was not penalized but often resulted in sketches which were difficult to read. The stronger candidates calculated \(g'(1)\) correctly, but very few represented the gradient of the function at \(x = 1\) correctly.

Question 6: Functions and Calculus Many candidates were able to calculate \(f(0)\)  although some had a problem with \({2^0}\). Very few were able to find both roots in part (b), even when they represented correctly both zeros in their sketch. Many sketches were reasonably well drawn, presenting a smooth curve with correct points of intersection. Many lost a mark for not labelling their axes and the given scale was ignored by some, which was not penalized but often resulted in sketches which were difficult to read. The stronger candidates calculated \(g'(1)\) correctly, but very few represented the gradient of the function at \(x = 1\) correctly.

Question 6: Functions and Calculus

Many candidates were able to calculate \(f(0)\)  although some had a problem with \({2^0}\). Very few were able to find both roots in part (b), even when they represented correctly both zeros in their sketch. Many sketches were reasonably well drawn, presenting a smooth curve with correct points of intersection. Many lost a mark for not labelling their axes and the given scale was ignored by some, which was not penalized but often resulted in sketches which were difficult to read. The stronger candidates calculated \(g'(1)\) correctly, but very few represented the gradient of the function at \(x = 1\) correctly.